J Multidis Res Rev 2019 Inno Journal of Multidisciplinary Volume 1: 2 Research and Reviews
Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post Contemporary Quantum Chemistry
Ichio Kikuchi International Research Center for Quantum Technology, Tokyo, Japan Akihito Kikuchi*
Abstract Article Information
A new framework in quantum chemistry has been proposed recently Article Type: Review Article Number: JMRR118 symbolic-numeric computation” by A. Kikuchi). It is based on the Received Date: 30 September, 2019 modern(“An approach technique to firstof principlescomputational electronic algebraic structure geometry, calculation viz. the by Accepted Date: 18 October, 2019 symbolic computation of polynomial systems. Although this framework Published Date: 25 October, 2019 belongs to molecular orbital theory, it fully adopts the symbolic method. The analytic integrals in the secular equations are approximated by the *Corresponding Author: Akihito Kikuchi, Independent polynomials. The indeterminate variables of polynomials represent Researcher, Member of International Research Center for the wave-functions and other parameters for the optimization, such as Quantum Technology, 2-17-7 Yaguchi, Ota-ku, Tokyo, Japan. Tel: +81-3-3759-6810; Email: akihito_kikuchi(at)gakushikai.jp the symbolic computation digests and decomposes the polynomials into aatomic tame positionsform of the and set contraction of equations, coefficients to which ofnumerical atomic orbitals.computations Then Citation: Kikuchi A, Kikuchi I (2019) Computational are easily applied. The key technique is Gröbner basis theory, by Algebraic Geometry and Quantum Mechanics: An Initiative which one can investigate the electronic structure by unraveling the toward Post Contemporary Quantum Chemistry. J Multidis Res Rev Vol: 1, Issu: 2 (47-79). demonstrate the featured result of this new theory. Next, we expound theentangled mathematical relations basics of the concerning involved variables. computational In this algebraic article, at geometry, first, we Copyright: © 2019 Kikuchi A. This is an open-access which are necessitated in our study. We will see how highly abstract ideas article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the problems in quantum mechanics. We solve simple problems in “quantum original author and source are credited. chemistryof polynomial in algebraic algebra variety”would be by applied means toof algebraicthe solution approach. of the definiteFinally, we review several topics related to polynomial computation, whereby we shall have an outlook for the future direction of the research. Keywords: Quantum mechanics, Algebraic geometry, Commutative algebra, Grönber basis, Primary ideal decomposition, Eigenvalue problem in quantum mechanics, Molecular orbital theory; Quantum chemistry, Quantum chemistry in algebraic variety, First principles electronic structure calculation, Symbolic computation, symbolic-numeric solving, Hartree-Fock theory, Taylor series, Polynomial approximation, Algebraic molecular orbital theory. Introduction Dear Readers, If you are researchers or students with the expertise of physics or chemistry, you might have heard of “algebraic geometry” or “commutative algebra”. Maybe you might have heard only of these words, and you might the department of mathematics, not in those of physics and chemistry. Younot havemight definite have heard ideas of about advanced them, regions because of theoretical these topics physics, are taught such asin super-string theory, matrix model, etc., where the researchers are seeking the secret of the universe by means of esoteric theories of mathematics with the motto algebraic geometry and quantum mechanics. And you might be desperate in imagining the required endurance to arrive at the foremost front of the study... However, algebraic geometry is originated from rather a primitive region of mathematics. In fact, it is an extension www.innovationinfo.org of analytic geometry which you must have learned in high- the Gröbner basis theory and triangulation of polynomial set. Let us review the exemplary computation of hydrogen of the word goes as follows: molecule in [2]. schools. According to Encyclopedia Britannica, the definition algebraic geometry, study of the geometric properties Let φ , , be the wavefunctions, the atomic charges, i Za Ra of solutions of polynomial equations, including solutions in and the atomic positions. The total energy functional of three dimensions beyond three... Hartree-Fock theory is given by It simply asserts that algebraic geometry is the study of ∆ 2 Z polynomial systems. And polynomial is ubiquitous in every Ω= φφ− + a ∑∑∫ dr ii() r ()r branch of physics. If you attend the lecture of elementary ia2 ||rR− a quantum mechanics, or you study quantum chemistry, you always encounter secular equations in order to compute ′′ 1 ′ φφφii(rr ) ( ) j (r ) φ j (r ) the energy spectrum. Such equations are actually given by + ∑ drdr 2∫ ||r − r′ the polynomial systems, although you solve them through ij, linear algebra. Indeed, linear algebra is so powerful that you ′′ 1 ′ φφφijji(rr ) ( ) (r ) φ (r ) have almost forgotten that you are laboring with polynomial − ∑∫ drdr algebraic equations. 2ij, ||r − r′ Be courageous! Let us have a small tour in the sea of ZZ + ab QUANTUM MECHANICS with the chart of ALGEBRAIC ∑ a,b ||RRa − b GEOMETRY. Your voyage shall never be in stormy and misty mare incognitum. Having chartered a cruise ship, − λ(dr φφ ()r (r) − δ) the “COMMUTATIVE ALGEBRA”, we sail from a celebrated ∑ ij,,∫ i j ij ij, seaport named “MOLECULAR ORBITAL THEORY”. The secular equation is derived from the stationary The molecular orbital theory [1] in the electronic structure condition with respect to the wave-function computation of quantum chemistry is computed by the solution of the secular equation, if one adopts the localized δΩ atomic orbital basis {φi} =0. positions with other optimizable variables ζ : δφi {Ri} { i} defined on the set of atoms at the The energy minimum with respect to the atomic coordinate gives the stable structure: H,({Rij} {ζζ})⋅Ψ = ER S,({ ij} { })⋅Ψ ,
Let us consider the simple hydrogen, as in Figure 1. in which the matrix elements are defined by H,ij = drφφi j ∫ Assume that the two hydrogen atoms are placed at RA
and RB . By the simplest model, we can choose the trial and 3 wave-functions for up-and down-spin at the point z ∈ as = φφ follows: S.ij ∫dr i j 1 In these expressions, is the Hamiltonian; the vector φ ( z) =( aexp( −+ zb) exp( − z) ) Ψ up π AB Ψ= χφ ∑ ii; E is the energy spectrum. The Fock matrix H and represents the coefficients in LCAO wave-function 1 the overlap matrix S are, in theory, computed symbolically φdown ( z) =( cexp( −+ zdAB) exp( − z) ) and represented by the analytic formulas with respect to π the atomic coordinates and other parameters included in Where the Gaussian- or Slater- type localized atomic basis; they are, in practice, given numerically and the equation is solved z= zR − by means of linear algebra. In contrast to this practice, it is AB//AB demonstrated by Kikuchi [2] that there is a possibility of and abcd,,, are the real variables to be determined. Let φ φ symbolic-numeric computation of molecular orbital theory, ev and ew to be Lagrange multipliers λij for up and down . Since which can go without linear algebra: the secular equation these two wave-functions are orthogonal in nature, due to is given by the analytic form and approximated by the the difference of spin, we assume that the multipliers are polynomial system, which is processed by the computational algebraic geometry. The symbolic computation reconstructs and decomposes the polynomial equations into a more tractable and simpler form, by which the numerical solution H H of the polynomial equation is applied for the purpose of obtaining the quantum eigenstates. The key technique is Figure 1: The image of a hydrogen molecule, composed of two atoms.
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12470*t=0 diagonal: λij = 0 for ij≠ . From these expressions, all of the integrals involved S[2]156*s*u^2*r^4-1068*s*u^2*r^3+2248*s*u^2*r^2- in the energy functional are analytically computed. (As 80*s*u^2*r for the analytic forms of the integrals, see the supplement ...... of [2].) Then, with respect to inter-atomic distance , RAB Taylor expansion of the energy functional is computed up +992*t*u*v*r^2-160*t*u*v*r+40*t*u*v=0 to degree four, at the center of RAB = 7/5. The polynomial S[3]156*s^2*u*r^4-1068*s^2*u*r^3+2248*s^2*u*r^2- approximation is given by 80*s^2*u*r OMEGA = (3571 - 1580*a^2 - 3075*a*b - 1580*b^2 - ...... 1580*c^2 +126*u*r^4-882*u*r^3+1748*u*r^2+1896*u*r- + 625*a^2*c^2 + 1243*a*b*c^2 + 620*b^2*c^2 - 12470*u=0 3075*c*d S[4]-52*s^2*v*r^4+236*s^2*v*r^3-32*s^2*v*r^2- + 1243*a^2*c*d + 2506*a*b*c*d + 1243*b^2*c*d - 104*s^2*v*r 1580*d^2 ...... -30*v*r^4+330*v*r^3-1148*v*r^2+760*v*r-170*v=0 ...... δΩ The stationary conditions δΩ , , with respect to , δ ev δ ew ...... , are the normalization condition for the wave-functions. - 86*a*b*c*d*r^4 - 17*b^2*c*d*r^4 + 12*d^2*r^4 - They yield two equations: 4*a^2*d^2*r^4 S[5]-13*s^2*r^4+139*s^2*r^3-458*s^2*r^2+63*s^2*r - 17*a*b*d^2*r^4 + 13*a*b*ev*r^4 + 13*c*d*ew*r^4)/1000 ......
where the inter-atomic distance RAB is represented by r -63*t^2*r-3986*t^2+1000=0 (for the convenience of polynomial processing). S[6]13*u^2*r^4-139*u^2*r^3+458*u^2*r^2-63*u^2*r The equations used in the study are quite lengthy, so we only show the part of them. We give the exact ones in the ...... appendix (supplementary material): the energy functional +63*v^2*r-14*v^2+1000=0 in Appendix A; the secular equations in Appendix B; the δΩ The stable condition for the molecular geometry Gröbner bases in Appendix C; the triangulation in Appendix D. δ r yields this: In order to reduce the computational cost, the numerical 312*s^2*u^2*r^3-1602*s^2*u^2*r^2+2248*s^2*u^2*r of decimal numbers. We make the change of variables from ...... coefficients abcd,,, to stuv are,, represented , in the following by the fraction,way: by the truncation ...... a=+=−=+=− t sb,, t sc u vd , u v +380*v^2+740*r^3-3903*r^2+7288*r-5102=0 Consequently, the wave-functions are represented by the For simplicity, however, we replace the last equation with sum of symmetric and anti-symmetric parts: t S[7] 5*r-7=0. φ (z) =(exp( −+ zz) exp( −)) a simple one (of the fixed inter-atomic distance) as follows: up π AB s By processing polynomials S[1],...,S[7], We obtain 18 +(exp( −−zz) exp( −)) It fixes the inter-atomic distance at 1.4. π AB polynomials in the Grönber basis J[1],...,J[18]. We only u φ = −+ − lengthy. The exact form is given in the appendix. down (z) (exp( zzAB) exp() ) show the skeletons of them, because the coefficients are too π J[1]=r-1.4 v +(exp( −−zz) exp( −)) J[2]=(...)*ew^6+(...)*ew^5+(...)*ew^4 π AB +(...)*ew^3+(...)*ew^2+(...)*ew+(...) δΩ δΩ δΩ δΩ The stationary conditions , , , , with respect to δ a δ c δ c δ d J[3]=(...)*ev+(...)*ew^5+(...)*ew^4+(...)*ew^3 t, s, u, v, yields those equations: -(...)*ew^2-(...)*ew+(...) S[1]32*s*u*v*r^4-336*s*u*v*r^3+992*s*u*v*r^2-160*s*u*v*r ...... J[4]=(...)*v*ew^4+(...)*v*ew^3+(...)*v*ew^2 +126*t*r^4-882*t*r^3+1748*t*r^2+1896*t*r- -(...)*v*ew-(...)*v
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J[5]=(...)*v^2-(...)*ew^5-(...)*ew^4-(...)*ew^3 _[2]=ew-(...) -(...)*ew^2+(...)*ew-(...) _[3]=ev-(...) J[6]=(...)*u*ew^4+(...)*u*ew^3+(...)*u*ew^2 _[4]=v^2-(...) +(...)*u*ew+(...)*u _[5]=u J[7]=(...)*u*v*ew^2+(...)*u*v*ew+(...)*u*v _[6]=t J[8]=(...)*u^2+(...)*ew^5+(...)*ew^4+(...)*ew^3 _[7]=(...)*s^2-(...) +(...)*ew^2-(...)*ew-(...) T[3]: J[9]=(...)*t*ew^4+(...)*t*ew^3+(...)*t*ew^2 _[1]=r-1.4 +(...)*t*ew+(...)*t _[2]=ew^2+(...)*ew+(...) J[10]=(...)*t*v*ew^3+(...)*t*v*ew^2+(...)*t*v*ew+(...)*t*v _[3]=ev-ew J[11]=(...)*t*u*ew^3+(...)*t*u*ew^2+(...)*t*u*ew+(...)*t*u _[4]=v^2-(...)*ew-(...) J[12]=(...)*t^2-(...)*ew^5-(...)*ew^4-(...)*ew^3 _[5]=u^2+(...)*ew+(...) +(...)*ew^2+(...)*ew-(...) _[6]=t^2+(...)*ew+(...) J[13]=(...)*s*ew^2+(...)*s*ew-(...) _[7]=s+(...)*t*u*v*ew+(...)*t*u*v *s-(...)*t*u*v*ew-(...)*t*u*v T[4]: J[14]=(...)*s*v*ew-(...)*s*v-t*u*ew^2-(...)*t*u*ew-(...)*t*u _[1]=r-1.4 J[15]=(...)*s*u*ew+(...)*s*u+(...)*t*v*ew^2 _[2]=ew+(...) +(...)*t*v*ew+(...)*t*v _[3]=ev+(...) J[16]=(...)*s*u*v+(...)*t*ew^3+(...)*t*ew^2 _[4]=v +(...)*t*ew+(...)*t _[5]=u^2-(...) J[17]=(...)*s*t-(...)*u*v*ew-(...)*u*v _[6]=t^2-(...) J[18]=(...)*s^2+(...)*ew^5+(...)*ew^4+(...)*ew^3 _[7]=s -(...)*ew^2-(...)*ew-(...) T[5]: The triangular decomposition to the Gröbner basis is _[1]=r-1.4 T[1],..., T[5]. Here the only the skeleton is presented, while _[2]=ew+(...) thecomputed, details which are given contains in fivethe decomposedappendix. Observe sets of equationsthat one _[3]=ev+(...) to the last, the seven variables are added one by one, with the _[4]=v^2-(...) orderdecomposed of r, ew, set ev, includes v, u, t, s, seven in the entries; arrangement from the of firsta triangle. entry Now we can solve the equation by determining the unknown _[5]=u variables one by one. As a result, the triangular decomposition _[6]=t^2-(...) yields four subsets of the solutions of equations: the possible _[7]=s T[1]: electronic configurations are exhausted, as is shown in Table 1. Table 1: This table shows the solutions for the secular equation after the _[1]=r-1.4 triangulation. The electron 1 and 2 lie in the up- and down- spin respectively; and the result exhausts the possible four congurations of the ground and the _[2]=(...)*ew+(...) excited states. _[3]=(...)*ev+(...) Solution 1 Solution 2 Solution 3 Solution 4 t -0.53391 0.00000 -0.53391 0.00000 _[4]=v s 0.00000 -1.42566 0.00000 -1.42566 _[5]=(...)*u^2-(...) u -0.53391 -0.53391 0.00000 0.00000 v 0.00000 0.00000 -1.42566 -1.42566 _[6]=t ev -0.62075 -0.01567 -0.62734 0.01884 ew -0.62075 -0.62734 -0.01567 0.01884 _[7]=(...)*s^2-(...) r 1.40000 1.40000 1.40000 1.40000 T[2]: The total energy -1.09624 -0.49115 -0.49115 0.15503 electron 1 symmetric asymmetric symmetric asymmetric _[1]=r-1.4 electron 2 symmetric symmetric asymmetric asymmetric
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This is one of the featured results in [2]. The author of that A homogeneous polynomial is a polynomial, in which work had demonstrated the procedure of the computation every non-zero monomial has the same degree. in a factual way, but he had not explained the underlying Example 4.1 mathematical theory so minutely. Consequently, it is x32+++ x y xy 23 y is a homogeneous polynomial of algebra and algebraic geometry because these theories are degree 3. beneficial for us to grasp some prerequisites of commutative still strange to a greater part of physicists and chemists. In x32++ xy z 2is not homogeneous. the following sections, we review concepts of commutative algebra and algebraic geometry, which are utilized in this Example 4.2 (Homogenization) A non-homogeneous sort of computation. Next, we learn about Grönbner bases. polynomial Px( 1,..., xn ) is homogenized by means of an additional variable x0 . 32 2 x yy z commutative algebra surrogate the eigenvalue problem of h =3 + + =++32 2 For Ptxyz(, , , ) t x xy zt, linearWe will algebra. find Then that thewe apply “primary our knowledge ideal decomposition” to solve simple in t tt t problems of molecular orbital theory from the standpoint of Ideal polynomial algebra. In the end, we take a look at the related Definition 4.1 An ideal I in the polynomial ring S is the set topics which shall enrich the molecular orbital theory of polynomials, which is closed under these operation: with a taste of polynomial algebra, such as “polynomial ∈ and then iI1 iI2 ∈ ii12+∈ I of these methods will show the course of future studies. iI1 ∈ and α ∈ S thenα ⋅∈iI. optimization” and “quantifier elimination”. The employment Basics of Commutative Algebra and Algebraic We usually denote an ideal by the generators, such as Geometry I= ( xy,) or J=( x22 + y, xy) . Our handy tool is polynomial and our chief concern is how to solve the set of polynomial equations. Such topics I+= J{ f + gf : ∈ I and g ∈ J } are the themes of commutative algebra. If we would like The sum and the product of the ideals are defined by to do with geometrical view, the task lies also in algebraic IJ={ f . g : f ∈∈ I and g J } geometry. From this reason, in this section, we shall review of polynomials. IJ:=∈∈∈ { f SfgI : . for all gJ } mathematical definitions and examples related to the theory The ideal quotient is defined to be the ideal N. B.: The readers should keep in mind that the chosen For two ideals I and m topics are mere “thumbnails” to the concepts of profound ∞ mathematics. As for proofs, the readers should study from Im::∞ = Imi . , the saturation is defined by more rigorous sources; for instance, for commutative i=1 algebra, the book by Eisenbud [3] or the Book by Reid [4]; (Observe that Im::i ⊂ Imfor i < j. In many cases, we for algebraic geometry, the book by Perrin [5], for Gröbner bases, the works by Cox, Little, and O’Shea [6,7], the book by ideal quotients, as the extension by union with Im: i shall Becker and Weispfenning [8], or the book by Ene and and do not have to consider the union of infinite i, if numberthe ring ofis Herzog [9]. Noetherian, as will be discussed later.) “saturates” and stop to grow at some finite Polynomial and ring The radical of an ideal I
I=∈∈{ f Sf :k I foris defined someinteger by k> 0} indeterminateA polynomial variables. should Let be K defined in a ring. A ring is composed by the coefficient (in some field) and the Affine algebraic set (Affine algebraic variety) polynomial ring S= Kxx[ 12, ,..., xn ] is the K-vector space, with the basis elements (monomials) be of thethe formcoefficient field. The Definition 4.2. Let S be the subset of a ring kx[ 1,..., xn ]. We
denote the common zeros of polynomials Px( 1,..., xn ) in S by xxcc12.... xcn (= Xc ) 12 n n VS( )=∈∀ { x k Px (1 ,...., xn ) ∈ S such that Px ( )= 0} a ∈ f= CXa with ci . The polynomial is represented by ∑ We call VS( ) (){XfC=a ≠ 0} and we use the notation sup a S C cc12 cn degree of a monomial X= xx12 .... xn by the affine algebraic set (or affine variety) 22 . We define the definedExample by 4.3 In [ XY, ] ,VX( + Y) ={ X =±−1 Y} . n C Example 4.4 In [ XY, ], VX(22+=== Y ) X Y 0. X = ∑ci i=1 Example 4.5 In kxx[ 12, ,..., xn ],V ({1}) = ∅ .
The degree of polynomial is given by n Example 4.6 In kxx[ 12, ,..., xn ],Vk({0}) = . deg(ff )= max{ XCc : X∈ supp( )} notable. Also, these properties of the affine algebraic set are J Multidis Res Rev 2019 51 www.innovationinfo.org
• If AB⊂ ,VB( ) ⊂ VA( ) . Several resources use the term “quotient ring” or “factor ring” for the same mathematical object. • A point aa= ( 1,..., an ) In general, an ideal depicts geometric objects, which Va( ) =−−( xa,..., xa) 1 n is an affine algebraic set: might be discrete points or might be connected and lie in
• If I= ( ff12, ,..., fn ) then several dimensions. They are represented by the residue class ring: VI( ) = Vf( 1 ) ∩∩... Vf( n )
• kxx[ 12, ,..., xn ] / I. algebraic set: The intersection of affine algebraic sets is also an affine Let us see several examples. VS= V S ( jj) Ji 2 Example 4.10 [xx] /2( − ) The elements in the ring Rx[ ] Or n are the polynomial f( x) =++ a01 ax axn . We divide fx( ) by VI( ) ∪= VJ( ) VIJVI( ) =∩( J) . x2 − 2 , so that =++ ⋅−2 • The f( x) b01 bx g( x) ( x 2) algebraic set: finite union of affine algebraic sets is also an affine The residue b01+ bxis the representative of fx( ) when it VI( ) ∪= VJ( ) VIJVI( ) =∩( J) . is mapped into the residue class ring. We might assume that ∞ (the representative of x in the residue class ring) would The interpretation of the saturation IJ: in algebraic x geometry is this: the saturation is the closure (in the sense of not be an undetermined variable, but a certain number α (outside of ) such thatα 2 = 2 . topology) of the complement of VJ( ) in VI( ) . 22 Definition 4.3 Let V be a subset of k n (where K is an Example 4.11 [xy,/] ( x+− y 1) . ideal of V We assume that the representatives x and y of x and y in the residue class ring would not be undetermined arbitrary field).= The ∈ is| ∀∈defined as = IV( ) { f k[x1 ,...,xn ] x V , f( x) 0 } variables, but a pair of numbersα and β such that αβ22+=1, and that ( xy, ) depicts a unit circle, as a one- In other words, IV( ) is the set of polynomial functions 2 which vanish on V . dimensional geometric object in . Example 4.12 Example 4.7 I(∅=) kX[ 1,..., Xn ]. [xy,] / x22+− y 1, x − y Example 4.8 For the ideal ( ) The representative x and y of x and y are considered 2 32 I=(( x − yx) ), to the points or (1/ 2,1/ 2) (which lie in the intersection of xy22+=1 and xy= ). This example is a simplest case of the saturation is given by zero-dimensional ideal. Ix:( )∞ =( x23 − y) . Prime and primary ideal There are two fundamental types of ideal: prime ideal 23 23 xy= is the composure of the curve xy= and the and primary ideal. doubled line x = 0 . The saturation Ix:( )∞ removes the = line x 0 from that composure; the point ( xy,) = ( 0,0) Definition 4.4 An ideal I ( ⊂ R ) is prime, if IR≠ , and, if (which lies on x = 0 ), however, is not removed, because the xy∈ I , then xI∈ or y∈ I. saturation is the closure. Definition 4.5 An ideal I is primary, if IR≠ , and, if xy∈ I , then I, denoted by n IV( ( J)), is computable. xI∈ or yI∈ for some n > 0 ; that is to say, every zero The ideal of an affine algebraic set of a ideal divisor of RI/ is nil-potent. 22 Example 4.9 For J=( Y,, Y) ⊂ [ XY], VJ( ) = {(0,0)} ⊂ and IV( ( J)) = ( XY, ) . Observe that the IV( ( J)) ≠ J. This Example 2.13 ( xx) [ ]is a prime ideal. is the consequence of the famous theorem of Hilbert 22 Example 4.14 P=( x,, xy y) ⊂ [ x , y] is a primary ideal: x (Nullstellensatz), which we will learn later. 2 and are zero divisors in RP/ ; xP2 ∈ and yp∈ . Residue class ring or quotient ring Example 4.15 Every prime ideal p is primary. IR⊂ be an ideal in a ring R, and f is an element in R. The set f+= I{ f + hh: ∈ I} is “theWe canresidue define class “residue of modulo class rings”. ”; Letis a representative of equivalent. f I f In the affine algebraic set, these two properties are the residue class fI+ . We denote the set of residue classes • Ideal I is a prime ideal. modulo I by RI/ . It also has the ring structure through addition and multiplication. • VI( ) is irreducible.
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The correspondence between variety and ideal subsets of X. The chain is the relation of inclusion as this:
In order to unify the algebraic and geometric views, let XX01 ... Xn us review the correspondence between varieties and ideals. We have seen that for a prime ideal P There are correspondences: set VP( ) kind of dimension related to prime ideals., the For affine a prime algebraic ideal n p , we canis an construct irreducible the chain closed of subset.prime ideals Hence with we definelength n a {Ideals of k [xx1 ,…, n ]}VX {Subsets ofk } of the form and . n pp01 ... ppn = {SubsetsX of kI}{Ideals of kx[ 1,,… xn ]} with prime ideals p0 ,..., pn . The chain is not necessarily where the ideal IX( ) and the variety VJ( ) are d in unique, and we denote the maximum of the length of chains the previous sections. by height ( p ). The Krull dimension of a ring is the efined A Also there are one-to-one correspondences: maximal length of the possible chains of prime ideals in A . We denote it by dim (A) n k {Radical ideals of kx[ 1 ,..., xn ]} ↔⊂{Varieties X k} Recall that for two prime ideals such that pq⊂ , ∪∪Vp( ) ⊃ Vq( ) ; the inclusion is reversed. n {Prime ideals of kx[ 1 ,..., xn ]} ↔⊂{Irreducible varieties X k}. Example 4.22 We have dimk ℝ[x_1,x_2,...,x_n]=n, because the maximal chain of primes ideals is given by 2 Example 4.16 For the ideal Ix=( ) ⊂ [ x], Jx= ( ) . We = haveVI( ) V( I) . (0) ⊂( x1) ⊂( xx 12,) ⊂⊂ ...( xx12 , ,..., xn ) . Example 4.17 For the non-prime ideal I=( xy) ⊂ [ x] , One often refers to the Krull dimension of the residue VI( ) = Vx( ) ∪ Vy( ) . class ring RI/ by “dimension of the ideal I”. The smallest prime ideal p ideal is I itself (when I is prime) or a minimal When we compute or analyze VI( ) for an ideal I , it might 0 p0 including I (as one might study a non-prime ideal I), and be convenient for us to work with VIor its irreducible p0 ( ) the dimension counting ascends from as the start-point. component Vp( i ) simpler. The principle of “divide and conquer” is equally Example 4.23 Consider I= ( xy) in R= [ xy, ]. This ideal is not effective in symbolicbecause computation. the defining ideal would be prime. In order to count the dimension of the ideal I , the chain of primes is given by ( x) ⊂ ( xy, ) or ( y) ⊂ ( xy, ) , since Noetherian ring ( x) and ( y) are the minimal prime ideals over ( xy) . Hence Definition 4.6 dim(RI )= 1. generated. A ring is Noetherian when it is finitely Example 4.24 Let f be a polynomial in the ring S of dimension n, (say, [xx12, ,..., xn ] ). We assume that f is not the zero divisor, ideals: if there is an ascending sequence of ideals, such that i.e. there is no element gS∈ such that gf⋅=0 and that it is This means that there is no infinite ascending sequence of not invertible, namely, there is no element gS∈ such that I11⊆⊆... Ik−+ ⊆ II kk ⊆ 1 ⊆... gf⋅=1. Then Rf/ ( ) has dimension n-1. A simple example is
it terminates somewhere in the following sense: [xy,/] ( x− y) yx= . = = . IInn+1 These examples, which seem is the to linebe trivial defined but by demand the equation us a certain amount of technical proof to show the validity of the Example 4.18 xx, ,..., x and [xx12, ,..., xn ] are Noeterian [ 12 n ] statements. (See the argument in [5].) rings. If there is a computational process in these rings which would create the ascending sequence of ideals, it must Zero-dimensional ideal
As we have seen, an ideal I in a ring R[x1,x2,...,xn] by the set of polynomials. The points (x ,x ,...,x ) R, which terminateExample after4.19 finiteIf R numberis a Noetherian of steps. ring, then RX[ ]is 1 2 n Noetherian (as the consequence of the Hilbert basis theorem) are the zero of the generating polynomials, determineis defined the [3]. By induction, RX[ ,..., X] is also a Noetherian ring. V(I ∈ V(I) is a 1 n discrete set, the ideal I is said to be “zero-dimensional”. If we Example 4.20 The power series ring Rx[ ] is a Noetherian haveaffine to algebraic solve the set set of). polynomials, If the affine wealgebraic have to set work with ring.
Example 4.21 If R is a Noetherian ring and I is a two-sided thezero-set zero-dimensional for general cases. ideal, Hencewhere itwe is shall important find the to solution foresee ideal (such that for aR∈ , aI⊂ I and Ia⊂ I ), then the aswhether the discrete an ideal set is of zero-dimensional points. It is fairly or difficult not. The to criterionfind the residue class ring is RI/ also Noetherian. In other words, the for an ideal to be zero-dimensional is given by Gröbner basis image of a surjective ring homomorphism of a Noetherian theory, which we shall see later. ring is Noetherian. Nullstellensatz Dimension and Krull dimension Assume that k Let be a topological space. The dimension of is the X X Theorem 4.1 Weak Nullstellensatz Let I k [x1,...xn] be an maximum of the lengths of the chains of irreducible closed is an algebracally closed field. ideal (different from k [x1,...xn] itself), then V(I) is nonempty. ⊂ J Multidis Res Rev 2019 53 www.innovationinfo.org
Theorem 4.2 (Nullstellensatz) Let I k [x ,...x ], then 1 n Example 4.25 [xx1,..., n ] and [xx1,..., n ] are integral domains. I(V(I I. ⊂ Definition 4.8 A unique factorization domain (UFD) is an Spectrum))=√ of ring and Zariski topology integral domain in which any nonzero element has a unique factorization by the irreducible elements in the ring. we can adopt a view of geometry. In this section, we see a bit ofAs it. the set of polynomials define the geometric objects, Example 4.26 is a UFD. We say that a variety is irreducible when it is not empty Example 4.27 F, F [x] is a UFD. and not the union of two proper sub-varieties, namely, Example 4.28 ForFor a afield UFD R, R[x] is a UFD. By induction, XX= ∪ X for X and X⇒= XX or XX = . 12 1 2 1 2 Rx[ 1,..., xn ]is a UFD. For a prime ideal P in a ring S, V(P) is irreducible. By the last example, [xx1,..., n ] and [xx1,..., m ]are UFDs. For a ring A Hence any polynomial in these rings has unique factorization. However there are a lot of example which is not a unique Spec(A) = {prime, we define ideals the of A}ring spectrum by factorization domain.
For A=K [x1,...xn]/J there is a one-to-one correspondence Example 4.29 R[ X,,, Y Z W] /( XY− ZW )is not a UFD, since between Spec( A ) and irreducible sub-varieties of V(J). it permits two different factorization for one element: Spec( A) ↔ {irreducible sub-varities VJ( ) }. a= XY = ZW . Completion: formal description of Taylor expansion Every maximal ideal (hence being prime) in A=K [x1,...xn]/I K (such as ) and an ideal In the computation of molecular orbital theory through I has the form computer algebra, as is presented in the introduction, we with an algebraically closed field approximate the energy function by polynomials. We simply xa−−,..., xa ( 1 n ) replace the transcendental functions in the energy functional (such as exp Ax or erf( Bx) with the corresponding formal for some point(a1,...,an V(I). (Notice that, in case of I= (0), ( ) V(I)=Kn. Hence there is a one-to-one correspondence power series, and we truncate these series at the certain ) ∈ degree. For this purpose, we execute Taylor expansion for ↔ VJ( ) m-Spec the variable at a certain center point in the atomic distance. by means of If we increase the maximum degree in Taylor expansion ( xa−,..., xa −↔) ( a,..., a). 11nn by the exact analytic energy functional. Such a circumstance The Zariski topology of a variety X couldtoward be the represented infinity, the in computation a formal language would of converge mathematics. to that that the sub-varieties Y X are the closed sets, whereby the union and the intersection of a familyis defined of variety by assuming are also ⊂ formal description of Taylor expansion. K, these two We need several definitions in order to present the statements are valid. For a ring S closed sets. For an algebraically closed field I would be written as follows: (i) Any decreasing chain n V1 V2 K , a “filtration” by the powers of a proper ideal eventually terminates. FS0( ::= S) ⊃ FS 12( =⊃ I) FS ⊃⊃ FSnn( : = I ) ⊃ ⊃ ⊃... of varieties in (ii) Hence any non-empty set in n has a minimal element. K The sequence is given by the inclusion relation. The I-adic topology. for closed subsets is called to be Noetherian. Observe that An “inverse system” is a set of algebraic objects ()Aj jJ∈ , which the Achain variety for which the Noetherian satisfies the varieties descending is descending, chain condition while filtration determines the Krull topology of is directed by the index J with an order ≤ . In the inverse system, we have a family of map (homomorphism), the Noetherian ring. the chain for the ideals is ascending when we have defined fA: ji→ A A): the Zariski ij topology of Spec(A), in which the closed sets are of the form between two objects, from the larger index to the We define another type of topology in Spec( for all i ≤ j VI( )=∈⊃ { P Spec( AP ) I } f =1 Two types of Zariski topology for varieties and Spec(A) smallerii i. The map satisfies have similar properties. The comparison is given in Chapter 5 and of the book of Reid [4]. fffik= ijο jk Unique factorization domain Let I= xx,,, … x and I=( xx,,, … x) . The FSii:= I is ( 12 i ≤ j ≤ kn ) 12 n ( ) Definition 4.7 An integral domain is a non-zero commutative the ideal in R, generated by the monomials of degree i in S. We ring in which the product of any two non-zero elements is also assume that the inclusion is given in the sense of ideal non-zero. so that ideals generated by monomials of lower degrees
J Multidis Res Rev 2019 54 www.innovationinfo.org should contain those generated by monomials of higher formal neighborhoods of X should be small enough so that i degrees. We set Ai = S/ FSby the quotient rings. Hence the Taylor series should be convergent. i entries in Ai = S/ FS nj+ i Consider the map from to monomials in FS A are represented by Example 4.31 [xx] / [ x] i j the set of polynomialsare up the to degreefinite polynomials,i −1. The operation in which of the [xx] / [ x] and the corresponding inverse system. Now ii i = map from S/ FSj toare S/ FSnullified,is to drop and the monomials of higher Sx= [ ]and FS( x)[ x] . A polynomial f in [x] has the open degrees and to shorten polynomials. In case of the polynomial basis of neighborhoods, which is given by the set of the form n approximation by means of Taylor expansion, the map is the fx+ [ x] .
The extension to the multivariate case in [xx12,,,… xn ] projectionThe inverse from finer system to coarser can approximations.be glued together as a is straightforward. mathematical construction, which is called the “inverse limit” (or “projective limit”). The inverse limit is denoted and We interpret the neighborhoods of 0 in the slightly different view. Any polynomial has the image in each of j [xx] / [ x] . Hence there are the different classes of polynomials around 0 , which are represented by nil-potent definedA=lim{() as Ai = a j j ∈∈ J∏ Ai a i = f ij() a j for all i≤ j } jJ∈ ε as follows, The inverse limit A has the natural projection π : AA→ 2 i {0+=aεε such that 0} (by which we can extract Ai ). ++εε23 ε = The inverse limit is a “completion” of the ring {0aa11 such that 0} S=[ xx,,, … x] I=( xx,,, … x) 12 n with respect to 12 n , when the likewise, inverse limit is taken for the quotient rings in the following n i n1+ way: {0+=∑ a1εε such that 0} i=1 ˆ = n SI lim ( S/ IS) . and so on. In other words, the choice of the neighborhoods ˆ of 0 is to choose the tolerable threshold, above which the SI is the ring of formal power series R[ xx12,,,… xn ] , and we can arbitrarily approximate the formal power series by monomials are admittedly zero. Localization someExample finite 4.30 polynomials Consider through exp( x) .the Let natural Sx= [projection.] and Ix= ( ) Let R= [ xx,..., ]be a ring of polynomial functions. We j 1 n . FS j is the set of polynomials of the form xfx( ). can consider the rational function (or the fraction) The Taylor expansion exp( x) up to degree j −1, 1++x( 1/2) x21 +( 1/( jx − 1) !) j− , lies in F/ FS j . In the fX( ) gX( ) inverse limit of Ai , namely, [x] , the formal power series 2 n of exp( x) ,1++x( 1/2) x + +( 1/ nx !) + for fg, ∈ R. This sort of fraction is determined locally We can assume the formal power series in R xx,,,… x on a subset X in n , such that gX( ) ≠ 0 is defined.[ 12 n ] of such that would represent the inverse limit of “glued” Taylor aa=( 1,..., an ) ∈ X S R expansions. In the algebraic formulation of molecular . For a fixed point , we define the subset orbital theory, by means of inverse limit, we can bundle the S=∈≠{ pa( ) Rx[ 1 ,..., xn ] | pa( ) 0} different level of polynomial approximation with respect to the maximum polynomial degrees. Hence the natural map Then, for the pair in (a,s), denoted (a,s), the fraction as/ means the model selection. set of fractions must be closed under multiplication and Such a mathematical formality might appear only to addition.can be well-defined, Moreover the although equivalence it is arelation “local function”.between two The complicate the matter in practice, but it is important to pairs in RS× is given by introduce a neat “topology” in theory. The topology is constructed as follows: an object (i.e. a polynomial) in a (a, s) ≡( a ', s ') ⇔− as ' a ' s = 0. ring has the nested (or concentrated) neighborhoods in S ; the open basis of the neighborhoods is generated by the powers of a proper ideal IS⊂ and represented as more general way. In commutative algebra, “localization” is defined in a Definition 4.9 Let R be a ring. x+∈ ISfor x S A subset SR⊂ is “multiplicatively closed” if 1∈ S and We say “open basis” in the sense of topology. If the reader ab∈ S for all ab, ∈ S. is unfamiliar with topology, simply image that polynomials around a polynomial x are sieved into different classes, For R and its multiplicatively closed subset S, the which are represented by the above form. The powers of equivalence relation between two elements in RxS is given the ideal I serve as the indicator of the distance between by polynomials. In the terminology of topology, the completion (a, s) ( a ', s ') ⇔ thereis an elementu∈ S such that u( as ' −= a ' s) 0 makes a “complete” topological space. If we consider the inverse system for Taylor expansions at a point X , the Then the set of equivalence classes
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We say x11= xIS +∈ is integral over R when it − a S1 R:,= a ∈∈ Rs S the following condition: s satisfies Definition 4.10 Let RS⊂ be a ring extension. An element is the localization of at R the multiplicatively closed subset sS∈ is integral over R S. It is a ring with addition and multiplication, equation dd−1 if it satisfies a monic polynomial a a' as ''+ a s s+ as ++ a =0 += 1 d s s'' ss with aRi ∈ for all id=1,..., . and If every element in S is integral over R , then S is a a'' aa integral over R . We say that S is the integral extension of R . ⋅= . s s'' ss Then these statements hold: Let be a prime ideal in a ring . As the Example 4.32 P R If s1,..., sSm ∈ are integral over R , Rs[ 1,..., sm ] is integral set S= RP\ is multiplicatively closed, we localize R by over R . S. It is usually denoted by RP and called the localization of R at the prime ideal P. This localization has the exactly The succession of integral extensions makes the integral extension. If S is integral over R and T is integral over S, then one maximum ideal PRP . In particular, for R= [ xx1,..., n ] and =∈= ∈n is T integral over. P{ f Rfa( ) 0for a point a } , RP is the set of rational a . Example 4.36 [ y] →−[ x,/ y] ( xy 1) is not an integral extension. From the geometrical viewpoint, the functionsExample well4.33 defined Let R aroundbe a commutative ring and let f be a non-nilpotent element in R . A multiplicatively closed correspondence by this map (by inverting the direction) subset S is given by{fnn |= 0,1,...}. The localization gives us the projection of the hyperbola xy =1 to y -axis. −−11 There is no point in xy =1which is projected to the point S R= R[ t] /1( ft −=)( R f ) . y = 0 in y -axis, while any other points in y -axis have a Example 4.34 Let X= V( xy) preimage in xy =1. by the ideal( xy) ). Consider the ring A( X) = [ x,/ y] ( xy) . For a point a = (1, 0 ) in x -axis, the function(an affine y algebraicis equal setto zero.defined So Example 4.37 Apply the change of coordinates to the y /1and 0/1 should be equivalent as the elements in SR(− 1) above example: xt→ and y→+ ts. Then we obtain 2 with S=∈≠{ f Rf |( a) 0} . Indeed xS∈ and x( y⋅−1 01 ⋅) = xy = 0 [s] →[ t,/ s] ( t +− ts 1) , which is an integral extension. From in AX( ) , although ( yy⋅−1 01 ⋅) = ≠ 0 in [xy, ]. Hence the the geometrical viewpoint, the correspondence between equivalence between y /1 and 0/1 is proved. (This argument two rings gives us the projection to the hyperbola tt( += s) 1 is not valid for a = (0,0) , because x = 0 at that point and it is to t tt( += s) 1, namely, not in S .) ((1/2)(ss±+2 4) , s) , which are projected to an arbitrary point s in s-axis.-axis. The One change always of coordinatesfinds two points with inpolynomial maps of Consider the secular equation Example 4.35 this sort (which might be non-linear) enables us to obtain an 01− xx integral extension of a ring, even if the domain of the map is = e . not an integral extension (Noether-normalization) [2]. −10 yy Example 4.38 Consider the ideal I=+( x ey, y +⊂ ex) [ x,, y e]. The roots are given by( xye, ,) =−−( tt , , 1) ,( t , t ,1) ,( 0,0,0) I represents a simple secular equation VI( ) by the ideal I=++( x ey, y ex)in R= [ xy, ]. Let P=( xye,, + 1) and = . Since − S RP\ 01 xx. , andis represented they lie in by the another affine algebraic basis set set (by a Gröbner basis = e I −10 yy with lexicographic order xye>>) as
2 I=( y( e −+1,) x ey) [xy,] → [ exy ,,] / I is not an integral extension. In fact, I can be represented by another basis (−+y ye2 , x + ye), but there it follows that is no monic equation for e VI( ) − which is projected to the non-zero vector ( xy, ) unless e takes I⋅ S1 R = y e ++1, x ey ( ( ) ) certain particular values .(the One cannoteigenvalues). find the This point example in because e −1 is not in P and it is an invertible element in is an application of integrality check by means of Gröbner −1 SR(− 1) . The ideal IS⋅ R basis. (This useful idea will be expounded later.) VI( ) looks like” locally around the point ( xye, ,) =( 0,0, − 1). Indeed, the eigenvalue e is the describes root of the “how determinant the affine algebraice2 −=10and set Normalization the parabola fe=2 −1 looks like ge=21( + ) locally around Let us consider a curve y2= xx 32 + in [xy, ], as in Figure e = −1. 2. The curve has a singularity at (0,0) where two branches Integrality intersect. Let t= yx/ . Then y= tx , y2= xx 32 + , and tx2 =( +1) Consider the extension of quotient ring . These equations make an ideal I=( y− tx, y2 − x 3 − xt 22,1 −− x ) in [xyt,,] V(I) has no singularity and = ⊂= 2 32 R Kx[ 21,... xnn] / I S Kx[ 1,... x] / I it maps to the curve y= xx + by the projection to [xy, ]. . The affine algebraic variety J Multidis Res Rev 2019 56 www.innovationinfo.org
with the normalization map S S . The algorithm by 1.5 Decker et al. enables us to compute such normalization. If ↪ I⊂= R Kxx[ 12, ,.... xn ] / Iis a radical ideal, the normalization by 1 the algorithm shall give the ideal decomposition of I . The
computation returns s polynomial rings RR1,..., s and s prime π :R→= Ri , 1,... s ideals IR11⊂⊂,..., IRsS and s maps { ii } such that 0.5 the induced map
π :Kx[ 1 ,..., xn] / I→ R11 / I ×× Rss / I, 0 whereby the target of π (the product of quotient rings) is the normalization. -0.5 As we shall see in section 6, the primary ideal decomposition is a substitution for the solution of eiganvalue -1 problem. It is not so surprising that we meet again the decomposition of an ideal in the desingularization of the algebraic variety. The secular equation is given as -1.5 -2 -1.5 -1 -0.5 0 0.5 1
{ axi11+ ax i 2 2 ++ axin n =ε xi i , = 1,... n} Figure 2: The curve y2 = xx 32 + with the singularity at (0; 0). or
This is an example of the resolution of singularity. This {0=fi ( x1 , x 2 ,..., xn) := ax i11 + ax i 2 2 ++ axin n −ε xi i , = 1,... n}. sort of procedure lies in a broader concept of “normalization”. ε which shall give the non-zero If S is an integral domain, its normalization S is the integral solution of the matrix equation closure of S S. Then we have to find
Definition 4.11in the (Normal) quotient Letfield S (thebe an field integral of fractions) domain of with a11 − ε a12 ax1n 1 0 K. Let f be any monic polynomial in Sx[ ]. If aa− ε a x0 every root ab/ ∈ Kof f also lies in S, then S is normal. 21 22 2n 2 = the field of fractions Example 4.39 One can prove that every UFD is normal. aan12n ann − ε xn 0 Definition 4.12 (Normalization) Let R be a commutative ring with identity. The normalization of R is the set of Observe that the matrix in the left-hand side is the R which satisfy some Jacobian matrix R . elements in the field of fractions of ∂f J = i monicExample polynomial 4.40 Observe with coefficientsthat, in the above in example, t is in the ij ∂ 232 x j S=[ xy,/] ( y −− x x) , and observe that the equation tx2 −−1is a monic polynomial which guarantees thatquotient t is integral field of over S. In fact, in order to prove that this { fi} is determined by the rank condition of the Jacobian resolution of singularity is literally the normalization, it is matrix,The namelysingular by locus the ofcondition the algebraic that the affine Jacobian set defined matrix by is necessary for us to do the argument in detail. So we omit it not of full rank. The desingularization is the normalization, now. and the latter would result in the decomposition of an ideal,
23 by the algorithm of Decker. Example 4.41 Consider Ixy=( − ) and VI( ) . By setting 3 xt= and yt= 2 , the coordinate ring Cx[ , y ]/ I Ct [32 , t ]. Let us 32 22 denote tt, by (V ) . Then t is a root of polynomial st− given in [10,11]. in (Vs)[ ] , and tV∉( ) . Hence (V ) is not normal. It is not so difficult to trace the algorithm of normalization A= Kx[ 1,..., xn ] / I. For If= ( ,..., f) The Jacobian ideal Example 4.42 In the above example of the resolution of 1 m Jac(I ) cc×Weminors need of some the matrix definitions. Let singularity, t is contained in the normalization of (V ) . As 22 is defined by the t is the root of s−∈ t( Vs)[ ] , every element of [t]is also a root of some monic polynomial in (Vs)[ ]. (To prove the ∂∂ff validity of this statement, we need some arguments by 11 ∂∂xx commutative algebra ) Hence, [t]in the normalization of 1 n (V ) . Besides, [t] is a UFD, hence normal. Therefore [t]is the normalization of (V ) . ∂∂ ffmm ∂∂xx1 n = S Kx[ 1,..., xn ] / I S= Ky[ 1,..., ym ] / I ' In fact, the procedure to find a normalization of is to find another ring J Multidis Res Rev 2019 57 www.innovationinfo.org
2 Where, cn= − dim( X) with X= VI. Then the singular 22y ( ) Te= = locus of A is given by sloc( AV) =( Jac( I) + I). x2 Then we execute the computation by following these We have the extended ring steps. A' = A[ T] /( e+ T , eT +− 1, T22 e ) Compute the singular locus . Let be a radical ideal such 22 • I J =[xyeT, , ,] /( x++−+ eyy , exy , Txe , TeT , +− 1, T e ) that VJ( ) contains this singular locus. with the ideal I'=++−+( xeyy , exyTxeTeT , , , +− 1, T22 e ) . This • Choose a non-zero divisor gJ∈ and compute gJ: J . For a ring A' is normal. Indeed, after some algebra (in fact, by homomorphism from J to J , denoted by Hom( JJ , ) , and means of computer algebra), we can check that the singular for a non-zero divisor x in J , xHom( J , J) = xJ : J . locus of VA( ') is an empty set. (We check the emptiness = by the computation of Gröbner basis, which should be • Let g01(: gg) , ,..., gs be the generators of gJ: J . There are quadratic relations of the form generated by(1) .) From this reason, for a radical ideal J which contains the singular locus, we have JA= '(the ring s ggg j ij ij itself). Hence HomJJ ,= A ' and the criterion for the ik= ζζ∈ . A' ( ) ∑ kk, with A ggk =0 g As T is integral over A and actually Te= ± in A , the • Also there are linear relations between gg,1 ,..., gs (the normality is satisfied. syzygy) of the form ideal I ' is represented in two ways by the substitution for T: s g 2 η k = Ia =( x + eyy, + exy , − exee ,, += 1) ( 1) ∑ k 0 k =0 g 22 • Then a surjective map is given by Ib =+( xeyyex, + ,1 −=+− e) ( xey,1 e ) g i These two ideal lie in [xye,,]. As I is trivial, we adopt AT,..., T Hom( J , J) , Ti a [ 1 s ] g Ib for the purpose of normalization. In addition, when we introduce the variable , we implicitly assume that ≠ . • The kernel of this map is an↠ ideal I1 generated by the T x 0 quadratic and linear relations of the form When x = 0 , the ideal I is given by ss I= I( x) = ( xy, ) . − ζηij 0 ∩ TTij∑∑ k T k, kk T kk=00= Then the normalization of A is given by two rings: and it yields the extension of A by [xye,,] / I0 = [ xye ,,] /( xy ,) and A11= AT[ ,..., Ts ] / I 1 [xye,,] / I=[ xye ,,] /( x +− ey ,1 e) • This ring A1 may be normal. If not, we make the extension of A1 Compare this result to the example of primary ideal we arrive at the normalization. decomposition with the same ideal I which we will compute again. After finite steps of the successive extensions, • The criterion of normality: A is normal if and only if in section 6. We observe that the normalization has done the decomposition of the ideal imperfectly. A= HomA ( JJ , ) have to add extra . Then the algorithm stops. .T Ifi this criterion is satisfied, we do not Hensel’s lemma Consider I=++( x ey, y ex) in [xye,,]. Consider the problem to solve the equation 2 Let A= [ xye,,] / I. The Jacobian matrix of I is fx( ) = x −=70
Let us substitute a0 =1 in fx( ) : 1 ey . =−≡ ex1 f (1) 6 0mod3 Let = + . Then Then as1 13 2 2 fs(a )≡− 6 + 6 mod 3 . Jac(I) =−−−( 1 e , x ey , xe y) , or more simply, and. 1 Hence, if s =1, then sloc( A) = V( Jac( I) += I) V(1 − e2 , xy , ) (Recall that the Gröbner basis 2 2 −+ 2 . of I is {x( e1,) y xe}). Hence we set J=(1 − e , xy , ) as a radical f (a1 )≡− 6 + 6.1 = 0mod3 ideal containing the singular locus. Choose x as the non- 2 Let as2 =+1313 ⋅+ . Then zero divisor in J . Then xJ= ( x2 , xy) , since xe(1− 2 ) vanishes 3 . as an element in A . Hence, xJ:, J= ( x y) . Let T= yx/ . Then fs(a2 )≡+ 9 72 mod3 there are two linear relations. Hence, if s =1, then
3 e+ T =0, eT += 1 0 f (a2 )≡+ 9 71.1 ≡ 0mod 3
and a quadratic relation Likewise, by setting aann= + s, we can determine
J Multidis Res Rev 2019 58 www.innovationinfo.org such that f (a )≡ 0mod3n+1 . It means that we obtain the 3-adic n and deg(qhh ) < deg ( 1 ) . solution of the equation 2 Example 4.44 fx( ) = x −+(1 t) , and Rt= [ ] . For 2 Xa=01 +⋅+⋅+33 a a 1 mt=( ) ⊂ R, we begin from Let us consider the polynomial fX( ) in [x]. We have p gx0 = +1 nnn′ +1 f( X+= pYnn) f( X) + pY f( X)mod p hx0 = +1 Hence if pfn ′( X) n+1 or, equivalently, if f ′(α ) 0 mod p (ll) ( ) 0 mod p , we obtain Y such that f( X+≡ pYnn) 0 mod p+1 by the and we obtain gh, in an iterative way. As usual, we fraction. ≢ simply write the roots of fx( ) = 0 by ±+1 t . Hensel’s lemma ≢ fX( ) implies the existence of a power series in [t] . Y = − n fX'( ) Real algebraic geometry Observe the similarity with Newton method in numerical The computation of hydrogen molecule, presented in the analysis. In other words, the p-adic solution of fx( ) = 0 as ∞ introduction, is done in the real number, and in the algebraic i ∑pYn . This is Hensel’s lemma, stated as follows. i=0 A= { p( xx , ,.. x) = 0 | i = 1,... r } Theorem 4.3 (Hensel’s lemma) If fx( )∈ p [ x]and a ∈ in12 p set defined by the polynomials: On the other hand, we can add extra constraint of the satisfiesfa( ) ≡ 0 mod p form and B={ qin( xx12 , ,.. x) ≥= 0 | i 1,... s } . fa′( ) p Hence it is important to study the existence of solutions then there is a unique α ∈ such that f α = 0 and of (A) with (B). ≢ 0 mod p ( ) α ≡ apmod . We review several results from real algebraic geometry In the lemma, there is a condition f ′(α ) 0 mod p , while, from now on. (As for rigorous theory, see the book by Bochnak et al. [12] or the book by Lassere [13].) from the above example, it seems that we should use fa′( n ) ≢ p for changeable an . In fact, by the construction, ≡= aaan 0 mod p, it follows that fa''( n ) ≡ fa( ) . Hence we variety: assume≢0 mod that condition only for the starting point a . These two statement are equivalent for an affine algebraic (i) the ideal IX( ) has real generators f11,..., fkn∈ [ xx ,..., ]. In commutative ring there is a related idea, called “Linear Hensel Lifting”, by the following theorem. (ii) XX= , (by complex conjugation)
Let R be a commutative ring. Let f, g0, h0 be univariate polynomials in R[x] and let m be a ideal in R. If f decomposes Definition 4.13 The set of real points Vf( ,..., f) with Then we define real affine algebraic variety and1 realk ideal. into the product of g and h in Rm/ , that is to say, 0 0 fin∈ [ xx1,..., ] f≡ ghmod m, 00 Definition 4.14 is Ancalled ideal a real I is affine called algebraic as a real variety. ideal, if it is and if there exist polynomials s and t in Rx[ ]such that property: , sg.00+≡ th . 1mod m generated by real generators22 and satisfies the following ai∈[ x12,..., x n] and a+ ... + a pi ∈⇒ I aI ∈ then, for every integer l we have polynomials g (l) and h(l) such that For any real algebraic variety, IX( ) is a real ideal. To see the difference between real and non-real ideals, consider( x) , (ll) ( ) l f≡ ghmod m, ( x2 ) , and (1+ x2 ) . The last two ideals are not real ideals. and Definition 4.15 Let p1,... prn∈ [ xx12 , ,..., x] polynomials. (ll) ( ) The set g00≡≡ gh, h mod m (l) (l) n The computation of g and h is done in a constructive way. Wp( 1,..., pr ) =∈≥≥ { a | pa11( ) 0,... pa( ) 0} gg(1) = (1) Let 0 and hh= 0 . And let is called as a basic closed semi-algrbraic set. A general (ll+1) ( ) l semi-algebraic set is the boolean combination of them. g= g + mqg and Theorem 4.5 radical as follows: h(ll+1) = h( ) + mql (Real NullStellensatz) Let us define the real h =∈22m +∈ ∈ ∈ We solve the equation I p[] xp∑ gjj Ifor some q [ xm ], \ 0 j (ll) ( ) f− gh Then it holds that = , where is the real +≡ I IV( ( I)) VI( ) gq00hg hq (l) mod m m = I. If J is a real ideal, J IV( ( J)) ; that is < so that we obtain qg and qh such that deg(qgg ) deg ( 1 ) to say, for a real ideal J, the radical ideal J (by the standard affine algebraic set of J Multidis Res Rev 2019 59 www.innovationinfo.org
1 h=− x2 ++ ax b . VJ ( ) ( ) J in ). D definition of commutative algebra) is equal to the ideal of 22 When D > 0 [x]. Example 4.45 For Ixy=( + ) , IV( ( I)) = ( xy, ) . (the affine algebraic set of Then it happens that gf+2 += h0 , where 1=f ∈ Monoid({1}), 22 2 Definition 4.16 gx∈∑[] ⊂Σ ({ 1,} these) , and polynomials h∈ I( x ++ ax bare) . Inwell other defined words, in the r 2 2 quadratic equation has no real root if and only if D > 0 . ∑∑[][]x=∈= p xp qxi ( ) for some r ∈, i=1 Example 4.48 Consider the secular equation (for a molecular orbital model of a simple diatomic molecule): 22ee1 r Σ=(g1 ,..., gn ) ∑∑σσ ee gg11... ∈[] x r 01− xx e∈{0,1} = e . −10 yy
ee12 em Mf(1 ,..., fm )= { f12 f ... f mm e 1 ,..., e∈ } The problem is equivalent to 2 =+= []x are the sums of squares of h1 x ey 0 Σ2 polynomials; ( gg1,... n )is the “quadratic module”, which is h2 =+= y ex 0. the Fromset of these polynomials definitions, generated by 2 and gg,..., ; M is []x 1 n Add the constraint: the multiplicative monoid generated by ff1,..., m . f=−≠ xy0 Theorem 4.6 (Positivestellensatz) Let {gkk |= 1,..., n }, A Gröbner basis (with respect to the lexicographic order {fii |= 1, .. m } , and {hll |= 1,... t } be the polynomials in =2 −+ [xx1,..., n ] . The following properties are equivalent. xye>>) of the ideal I= ( hh12, ) is H( ye y, x ye) . After some algebra, f 2 reduces to 22ye22+ y with respect to H . gxk ( ) ≥= 0, k 1,...,n Hence we have obtained (i) x∈n fx( ) ≠= 0, i 1,..., m i 2 =−+2 gf+ += h0 for h∈ Ihh( 12, ) and g21( ey) hxi ( )= 0, l = 1,..., t 2 . If e ≤−1, g=−+( 21( ey) ) 2 Positivstellensatz. Hence there is no real solution to the (ii) ∃g ∈Σ gg1,... n , ∃∈fM( ff ,... ) , and ∃∈h Ih( 1,... hl ) such that gf+ += h . problem. (Or we say thatand the non-symmetric it satisfies the wave condition function of ( xy, ) such that xy≠ is not the ground state of the molecule From this theorem, one can derive the following theorem, at eigenvalue −1.) also. Noether normalization Theorem 4.7 Let f and {gik |= 1,..., n } be polynomials in [x] Let RS⊂ be a ring extension. Remember how can be an and K=∈∈{ x | x Wg( 1,.., gn )}. Then the following statements hold. element sS∈ integral over R . 2 An element sS∈ is integral over if it (i) ∀∈x Kf,0( x) ≥ if and only if ∃p ∈, ∃ gh , ∈Σ ( g1 ,... gn ) Definition 4.17 R such that fg= f2 p + h . {rR∈ } ∀∈ > 2 i (ii) x Kf,0( x) if and only if ∃p ∈, ∃ gh , ∈Σ ( g1 ,... gn ) satisfies a monic polynomial equation with coefficients such that fg=1 + h dd−1 s+ rs1 ++= rd 0 Example 4.46 The equation is called an integral equation for S over R. If The theorem asserts that every element sS∈ is integral over R, we say that S is integral over R. We also say that RS⊂ is an integral extension. f > 0 Definition 4.18 Let S S= Kx[ ,..., x] / I. Then if and only if 1 n there are elements y1,..., ySd ∈ with the following properties. qf22= p ++ p 2, be an affine ring 1 n (i) yy1,..., d are algebraically independent over K.
for some polynomials qp,1 ,..., pn . (From the theorem, (ii) Ky[ 1,..., yd ] ⊂ Sis an integral ring extension. the set {x∈− | fx( ) ≥ 0, fx( ) ≠ 0} is empty if and only if 2 2 22 p If the elements yy1,..., d satisfy these two conditions, the ∃u = s11 + + sn− − fq ∈Σ( − f ) , ∃∈p , ∃=vf ∈M( f ) , such that 2 p inclusion Ky[ ,..., y] ⊂ Sis a Noether normalization for S . uv+=0 . Now we can choose ps11=,..., pn−− 1 = s nn 1 , p = f.) 1 d In other words, every non-negative polynomial is a sum of Example 4.49 Let I=+( x ey, y +⊂ ex) [ x,, y e], and let squares of rational functions. S= [ xye,,] / I. The residuals xy, (∈ S) are not integral over Example 4.47 Consider x2 + ax += b 0 . [e]. But the change of the variables
D , f , g , h to be x→ xy, →= y '4 x + ye , →+ 5 x 6 y + e Dba= − 2 /4 makes the change in I as follows: Define , 2 1 a , I→= I′ (20 x + 29 xy + 4 xe + x gx= + D 2 +6y22 + ye ,5 x + 6 xy +++ xe 4 x y ) f =1
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Then there arises a question: under what circumstance We can prove the residuals xy,'('∈= S[ xy ,','/' e] I) are one can express the solution of a differential equation integral over [e']. Indeed, after a hard algebra (by means of Gröbner basis theory and the variable elimination as we using exponents, integration, and also by adding algebraic I ' contains following elements? We proceed step by step, by adding more two polynomials elements which are constructed over the elements already learn later), we affirm that the ideal presented in the computation. The analytic solution given 65y3+ 28 y 222 e ++ 2 y 3 ye − 3 y by this way is called Liouvillian solution, although it is not always possible to construct it. The solvability condition is and given by the differential Galois theory [14-16]. 325x3− 38 x 2 e − 12 x 22 +− xe x In the application of eigenvalue problem in quantum mechanics, we can consider the one-dimensional Example 4.50 There are several types of the variable change Schrödinger equation: which conduct Noether normalization. ψ″= −λψ →=+λ ( x) ( Px2n ( ) ) ( x) • xi yx i i nn x with λi K , when K with the even-degree monic polynomial potential i−1 in the coefficient field 2 • x→ y = x − xr for2 ≤≤ in 21i− in −−1 1 i iis an ii infinite field. P( z) =+=++ z2n az in z bz i cz i r . 2n∑ i ∑∑ ii i=0 ii= 00= . We have to find an integer To do with the latter type is, in fact, the proof of the In the last representation of P2n , the polynomial is given existence of Noether normalization. Let fx( 1,..., xn ) be a by completing squares. Let us write the solution in the I . By the change of variables, following form: we obtain ψ ( x) = Psexp( ± fx( )) polynomial defining the idealn−1 ++rr. fxy( 12, x 1 ,..., yn x1) as the product of a monic polynomial Ps and
If the monomial of the highest degree with respect to + x1 xn n−1 bxk 1 is originally given by fx( ) = + ∑ k nk++11k =0 aa12 an Ax12 x xn it becomes {bi} and {ci} in the polynomial potential, the eigenvalue λ n i−1 Now we can establish the relation among the coefficients a1 + r . and Ps . The relation of this sort is given by a second-order Ax11∏( yi x ) = i 2 differential equation for Ps . As the consequence of differential Hence, after the change of the variables, the term of the Galois theory, the equation has solutions at the particular highest degree with respect to x1 is given by setting of {bi} , {ci} and λ . To solve the problem, one can
21n− utilize Gröbner basis theory and the variable elimination, a12+ ar ++ arn Ax1 which shall be explained later. If the equation is solvable, we If r is large enough, this term has the degree larger can obtain the analytic solution [16]. than any other monomials. Therefore x1 is integral over Other important concepts of commutative algebra Ky[ ,..., y] 2 n It is advisable that the readers should consult textbooks of the variety. Let I∈ Kx[ 1,..., xn ] be a proper ideal. Let A= VI( ) and grasp the concepts of more advanced technical terms, . If K y,.., y.] ∈There Kx[ ,..., is xanother / I is away Noether to define normalization, the dimension the 11dn such as “module”, “free module”, ”regular local rings”, dimension of A is given by the number d , namely, dim( A) = d so. Maybe the concepts of homological algebra would be “valuation”, “Artinian ring”, “affine algebraic variety”, or Differential Galois theory necessary, such as “functor”, “exact sequence”, “resolution”, “projective”, “injective”, “Betti-number”, and so on. Indeed, One of the important ideas concerning algebraic the research articles are frequented by these concepts. geometry, differential algebra, and quantum mechanics is the differential Galois theory. Gröbner Basis Theory Let us solve The Gröbner basis theory is the key technique of commutative algebra and algebraic geometry[17-19]. The yy'0−= We get . namesake is his advisor Wolfgang Gröbner. Ctexp( ) idea was first proposed by Bruno Buchberger, and the Let us solve The monomial orders y″+2'0 ty = In one-variable case, we implicitly use the “monomial order” through the ordering of degrees: 1<
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b b b xa (= xxa1 ...an ) = 1 m Let 1 n and x( xx1 ...m ) be the monomials. And with the monomial order < . The ”initial monomial” in< (f ) let a and b be sequence of superscripts (aa12, ,..., an ) and is the largest monomial included in f (from which the (bb12, ,..., bn ) . lc( f ) is the . Hence the “leading term” is given by the Definition 5.1 The lexicographic order. in< (f ) coefficient is removed). The “initial coefficient” ab . coefficient of xx< ⇔∃ iinaba(1 ≤ ≤ ) :n =1 ,..., i−− 11 = bab iii , < lc()in() f< f in< (0)= 0 . product of the initial coefficient and the initial monomial: The initial ideal is the monomial ideal, generated . We use anin
ab Thisxx< definition, if the right-mostis equivalent component to the following of a-b is statement. positive. Definition 5.3 The degree reverse lexicographic order: DefinitionThe Gröbner 5.4 Let basis I beis definedthe ideal now. in the polynomial ring R , with the monomial order < . The Gröbner basis is the sequence a Let deg(x )=++ a1n ... a of elements gg12, ,..., gm , such that in<< (Ig )= (in (1 ),...,in < ( gn ) . ab xx<⇔ We could represent the polynomial f by means of a ab sequence of polynomials uu, ,..., u∈ I in the following (1) deg(x )< deg(x ) 12 m ( ) way (the standard expression) or f=++ pu11 pu 2 2 pmm u + r (2) deg(xab )= deg(x ) and ∃i (1≤≤in) : such that ann= b,..., a i++11 = bab iii , > . (The right-most component of a-b is positive. ) (i) No monomial in the remainder is contained in the
3 ideal (in<< (uu1 ),...in (n ) ). Example 5.1 Compute (xyz+++ 1) in [xyz,,] . The monomials can be sorted by the different types of monomial (ii) in<< (f )≥ in ( puii )for all i order. (Here the degree of a monomial is given by this If r = 0 we say that f reduces to zero with respect to correspondence: xyz>>. uu12, ,..., um . In general case, the standard expression is not The lexicographic order: xyz>>; unique. From this reason, we have to use Gröbner basis, since the standard expression by Gröbner basis is unique x32+3 x y + 3 x 2 z ++ 33 x 2 xy 2 + 6 xyz ++ 63 xy xz2 + 6 xz and any polynomial reduces uniquely by this basis. For example, consider the case with f= xy − y2 , g= xy − , and +++3xy3 3 yzy 2 + 3 2 + 3 yz 2 + 6 yzyz +++++ 332 3 z 31 z 1 gx2 = in the lexicographic order. There are two standard = − 2 The reverse lexicographic order: z > y > x; expression: f= yg1 and f yg2 y . However, the ideal ( gg12, ) is generated by the Gröbner basis ( xy, ) , by which f z 3 +3yz2 +3xz2 +3z2 +3y2 z+6xyz+6yz+3x2 reduces to zero and it has the unique standard expression. z+6xz+3z+y3+3xy2+3y2+3x2 y+6xy+3y+x3+3x2+3x+1 Buchberger’s algorithm The degree reverse lexicographic order: >>; xyz The Buchberger’s algorithm is the standard way to x32+3 x y + 3 xy 232 ++ y 36333 x z + xyz + y2 z + xz 2 + yz 23 + z generate Gröbner basis [3,6,7,9,20-22] +3x22 + 6 xy + 3 y + 6 xz + 6 yzz + 3 2 ++++ 3 x 3 yz 31 f and g lcm(in ( fgl ),in ( )) cm(in ( fg ),in ( )) The difference between the lexicographic order and the Letspoly us(,) define f g = the S-polynomial<< off two− polynomials<< g cin ( f ) dgin ( ) reverse lexicographic order is apparent; it is simply a << reversal. The difference between lexicographical order and where c and d f and g . the degree reverse is subtle; it could be understandable by One can prove that these phrases by Ene and Herzog [9], are the leading coefficients of gg1,..., m are the Grbner basis of ideal I with respect to ...in the lexicographic order, > uv a monomial order < , if and only if u has “more from the beginning” ̈ if and only if than v; in the degree reverse lexicographic order, uv> spoly( gij, g ) reduces to zero with respect to gg1,..., m for if and only if u has “less from the end” than v... ij< . Initial ideal The computational step is as follows. = Let f ≠ 0 be a polynomial in the ring R Kxx[ 12, ,..., xn ] Step–0 Let G be the generating set of ideal I .
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Step-1 For every pair of polynomials pq, in the ideal , G • for all ij≠ , the monomials in g j are not divisible by compute spoly( p, q) and its reminder with respect to . r G in< (gi ) . Step-2 If all spoly( p, q)reduce to zero with respect to G , we Gröbner basis of zero-dimensional ideal have already obtained the Gröbner basis. If there are non- zero reminders r ≠ 0 , add r to G and repeat again at Step–1. The following statements are equivalent for a zero-
dimensional ideal I⊂= S Kxx[ 12, ,..., xn ] with a monomial order < . Let us see examples. The computation terminates after finite steps. 1. I is a zero-dimensional ideal. Example 5.2 Consider f= x + ye and R= [ xye,,] in R= [ xye,,], 2. VI( ) with respect to lexicographic order xye>>. In the beginning, 2 3. If G is a Gröbner basis, then, for any n, there exists G= { fg, }. As in< (fx ) = and in< (g ) = xe , spoly( f, g) = ef − g =−− y ye . 1≤≤i The affine algebraic set ν is finite. 2 gG∈ such that = j for some nu ≥ 0 . The leading monomial in< (spoly ( f , g )) is ye . As the in< ( gx) i i monomials in Spoly (f, g) is not included by the monomial 4. I S . module (in<< (f ),in ( g ))= ( x , xe ) , spoly( f, g) does not 5. Let Mon ( A) be the set of monomials in A . The set reduce to zero; indeed it is the reminder itself. We add is contained in finitely many maximal ideals of Mon(S) \ Mon( in< ( I ) ) h:,= spoly( f g ) to G so that G= { f,, gh} . Then we compute more of s-polynomials : spolyhg( , ) =−=− xy ye2 yf and 6. SI/ is a K is a finite set. spolyhf( , ) =−− xy ye23 =−− yf yeh . Since spoly( h, g ) and The statement-vector 3) enables space of us finite to detect dimension. a zero-dimensional spoly( f, g ) reduces to zero with respect to G , we obtain ideal from its Gröbner basis, if the latter contains polynomials ν the Gröbner basis G . In fact, the initial term of g= xe + y is i which have initial terms such that xi for any 1 ≤≤in. divisible that of f= x + ey , and h= ef − g , g is a redundant The feature of zero-dimensional ideal, given by statement term which we can remove from the basis safely. 5) and 6), will be useful for solving polynomial equations by Consider , = + 22 Stickelberger’s theorem, as is explained in section 5.12. Example 5.3 f1 = x + ey f2 y ex fxy3 =+−1 in R= xye,, , with respect to lexicographic orer xye>> 2 22 [ ] Example 5.5 For I=( x + y + z −1, xyzx ++ , ) ⊂ Rxyz [ , , ] , . We have G= { fff,,} . We compute the s-polynomials: 123 the Gröbner basis with respect to lexicographic monomial 2 2 spoly f, f= ye − y spoly f,1 f= xye −+ y 2 ( 12) , ( 13) , and order xyz>> is (2z−+ 1, y zx , ). This is the example of zero- 2 spoly( f23, f) =−+ xy y e e . These three polynomials reduce 2 2 2 to f4 = ye − y , f =-2y +1, and f=−+2 ye e. We get 5 6 For =2 + 22 + − ++ ⊂ , the G= { ffffff,,,,,}and we compute the s-polynomials and dimensionalExample 5.6 ideals andI( it x satisfies y z the1, x statement y z) [3). xyz,, ] 123456 Gröbner basis with respect to lexicographic monomial order the reminders. The only non-zero reminder is (1 / 2)(e2 − 1) xyz>> is (2y22+ 2 yz + 2 z − 1, x ++ y z) . As the ideal depicts the , to which spoly( f, f ) and spoly( f, f ) reduce. We have 45 46 intersection of the unit sphere and a plane surface, it is not G= fffffff,,,,,,= e2 − 1. We compute the { 1234567 } zero-dimensional. Although the Gröbner basis has the terms s-polynomials and ascertain that all of them reduce to zero. z2 and z , these terms are not initial terms. Hence this ideal Thus G={ xeyyexx +, + ,222222 + y − 1, xyey − +− 1, 2 y +− 1, 2 yeee + , − 1} is the does not satisfy the statement 6). y+ ex 22 xye−+ y2 1 Gröbner basis. However, , xy+−1, and 2 22 2 Example 5.7 For I=( x + y + z −1, xyzx ++ , ) ⊂ Rxyz [ , , ] , −+2ye e are redundant: indeed they have the initial the Gröbner basis with respect to lexicographic monomial monomials divisible by those of other polynomials order xyz>> is (2z2 −+ 1, y zx , ). This is the example of zero- (Gˆ =+−+{ x ey , 2 y221, e − 1)} and reduce to zero with respect to Gˆ . Thus we can chose Gˆ as the Gröbner basis, and indeed this dimensionalSyzygy ideals and it satisfies the statement 3).
For a Gröbner basis{ ff12, ,..., fm} , there are relations as Exampleideal satisfies 5.4 the Considerrequired property.Ix = ( + 1, 1) in [x]. As follows + = spoly( x+=1, x) 1, The Gröbner basis is ( xx1, ,1) {1} . This s Gröbner basis is never to be zero, and the set of equation = ∑sfii 0 xx+=10 ∧ = 0does not have any solution. In other i=1
by means of the set of polynomials{ss12, ,..., sm} . (A trivial VI = V ({1}) = ∅ . Likewise, in the more complicated ( ) example is ffi j−= f ji f 0 ) Such a set of polynomials is a case,words, by thecomputing affine algebraicthe Gröbner set bases of thisG, we ideal can check is empty the “module” (which is actually a vector space), and we call it the existence of the solutions of the set of polynomial equations. Syz( f12,,, f… fn ) . The basis set of this module is computed from the Gröbner basis. In the above algorithm the generated Gröbner basis is not first syzygy and denote it by properly “reduced” by the terminology of several contexts. A When the computation of Gröbner basis is completed, reduced Gröbner basis ( gg1,..., n ) should have the following there are relations among the generators of the basis of the property [23]: form:
• gi is 1. spoly( fi, f j) = q ij,1 f 1 + q ij ,2 f 2 ++ qij, m f m . The leading coefficient of each J Multidis Res Rev 2019 63 www.innovationinfo.org
reduction of Macauley matrix) [17,29,31,32]. For the case of homogeneous polynomial ideals, the complexity in the r= spoly f, f − q f − q f −− q f Letij us define( i j) ij,1 1 ij ,2 2 ij, m m Gaussian elimination method is bounded by
Then {rij } generates Syz( f,,, f… f ) . Moreover,even 12 n ω if pp, ,..., p is not a Gröbner basis, we can compute nD+−1 12 n O mD , D Syz( p12,,, p… pn ) through its Gröbner basis. Example 5.8 Let us compute the syzygy of where ω is a constant, m is the number of polynomials 22 {p123=+ txeyp,, =+ txeyp =+− x y 1} with the lexicographic [33], n is the dimension of the ring, and D is the maximum monomial order txye>>>. Syz( p123, p, p ) are generated by three trivial generators F4 anddegree F of the polynomials [33]. InF order [34, to 35] improve outperforms efficiency, the −−p,0, p , pp , ,0 , 0, − pp , , 5 5 ( 3 1) ( 21) ( 31) rowone canreduction employ computation more refined of methods, Macaulay such matrix as Faugère’s [33]. and by the non-trivial one, . Indeed the efficiency of ( xyeyeyex2+ 3 −, −2 − y 2 +− 1, ye 22) . on the chosen monomial order. The lexicographic order is convenientIn fact, the efficiency for theoretical of the algorithm study, but is highly it consumes dependent a Observe that the inner product of those generators and considerable quantity of computational resource. Thus one ( ppp,,) is zero. The generators form a vector space. 123 often has to compute the Gröbner basis by other monomial Sygyzy is a module, in other words, a kind of vector orders (say, the degree reversible lexicographic order) in space generated by the vectors, the entries of which are order to facilitate the computation, and then, one can remake polynomials. We can compute the second syzygy in the the computed result in lexicographic order, by means of syzygy, too. The successive computation of higher syzygy There is another problem in the Gröbner bases generation. vector generators of the first syzygy and, likewise, the higher FGLM (Faugère, Gianni, Lazard, Mora) algorithm [36]. enables us to construct the “resolution” of a module M in a which is apparent in practice. The Buchberger’s algorithm Noetherian ring; the computation terminates after a certain applies the operation of addition, subtraction, multiplication, and division to the polynomial system. It often causes a great step [23]. discrepancy in the degrees of the generated polynomials and step so that we do not find any non-trivial sygyzy in the last Church-Rosser property computation presented in the introduction of this article, one The reduction is a binary relation from one object to canthe observenumerical such scale a tendency. of coefficients Thus, forin the the finalpractical result. purpose, In the another, like an arrow going in one direction. We often call it one must utilize several tricks to keep the polynomials as the rewriting process. “slim” as one can [37-39]. Definition 5.5 A binary relation (denoted by the symbol Gröbner basis for modules → ) has the Church-Rosser property, if, whenever P and Q Let us review what is the module. A R − module M is an are connected by a path of arrows, P and Q have a common abelian group with scalar multiplication RM×→ M, such as destination R by the relation. (a, m) →∈ am M . It has the following property. • 1mm= A basis is a Gröbner basis if and only if the reduction with = respectHence to wethe canbases define has the Gröbner Church-Rosser bases in theproperty, other way:as is • (ab) m a( bm) illustrated in Figure 3.
Complexity of the Gröbner basis algorithm P Q The original algorithm by Buchberger, as is presented pairs of polynomials( pq, ), since such pairs give birth to R1 R2 s-polynomialshere, is not spoly so efficient.( p, q) , which It produces immediately a lot ofreduce useless to zero or produce redundant polynomials. There are a lot of P Q studies about the upper bound of the complexity of Gröbner bases, such as Hermann bound [24], Dube bound [25], Wiesinger’s theorem [26], and so on. Those bounds are determined by several factors: (1) the number of variables, (2) the number of polynomials in the ideal, (3) the number of possible s-polynomials, (4) the maximum degrees of the polynomials, etc. In the worst case, the complexity is doubly R exponential in the number of variables [25,27-30]. However, Figure 3: Two types of reductions by binary relations. (Upper) not Church- the computation of the Gröbner basis is equivalent to the Rosser type; (Lower) Church-Rosser type. If the basis of an ideal is not Gröbner, the reduction goes in several destinations as in the upper figure. On Gaussian elimination process of a large matrix (by the row the other hand, if the basis is Gröbner, the reduction is unique.
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• (a+=+ b) m am bm Example 5.10 Intersection of ideal I and J in R. Let y be an extra variable. IJ∩ is computable by the relation • a( m+= n) am + bm . I∩ J =( I ⋅⋅ yR[ y] +⋅ J(1− y) ⋅ Ry[ ]) ∩ R. The intersection of the right-hand side is computed by the elimination of variable y . A free module is the module with a basis set {ei}. We can compute Gröbner bases of free modules. One of the purpose Example 5.11 The ideal quotient I : J is computable. If s IJ:= (: I f ) of this sort of computations is to study the linear equation of Jf= ( 1,..., fs ) , then j . In addition, for a single j=1 polynomials: polynomial f, I∩=⋅( f) f( If: ) . We can compute If∩( ) and
obtain the generators{gg12,,,… gm} . Then If: is generated
aa11 12 a1n x 1 b 1 by{g12/ fg , / f ,..., gm / f} . Hence If: j is computed as the intersection of . aa a x b If: j 21 22 2n 2 = 2 Example 5.12 I and a aa a x b polynomial f in a ring S as follows. n12 n nn n n Saturation: it is defined for an ideal I: f∞ =∈>∈{ g S: thereexists i 0such that fgi I} for which one might ask for the linear dependence of Let be a new variable, and let I be the ideal generated rows. t in St[ ] by I and by the polynomial 1− ft . Then the saturation If: ∞ = I ∩ S. ue> ve if < or = and Example 5.13 Radical membership: f∈ I ⇔∈ fIi We define the monomial orderij in modulesij in theseij ways. uv> . for some i >⇔0 For all gS∈ , there exists i (> 0) such Position over coefficient : that fgi ∈ I. ⇔=If: ∞ S. Hence, if the ideal, Iˆ (Example: xe> xe ). 21 12 example 5.12, has the Gröbner basis {1} , then fI∈ . > : if > or = and defined in ueij ve uv vv Gröbner basis algorithm in a different light ij< . Coefficient over position The Buchberger’s algorithm is actually the elimination (Example: xe12> xe 21). In the similar way as in the case of polynomial, we chose reduction in the large matrix. Let us see how it would work. the leading terms of the elements in a module; we also in large matrices. It is the way of Faguère to do the row Consider the same problem: f1 = x + ye, f2 = y + xe, compute the s-polynomials and the reminders in order that =+−22 fxy3 1. We compute the s-polynomials ef1 − f2 and we obtain the Gröbner basis. The slight difference is that xf− f when = and we use the s-polynomial 13 in< (f ) uei in< (g ) = vei the matrix. as follows: x2 . The coefficients in these polynomials are given in xye xf1 110000 0 lcm( uv , ) l cm( uv , ) xe ef 001100 0 spoly(,) f g =f − 1 2 = ye lc() f u lc () g u f 001001 0 2 y2 f3 100010− 1 (Here we use the notation lc() x to represent the y 1 in< (x ) .) Application of Gröbner basis The matrix in the right hand side is the so-called Macaulay coefficient of matrix of the second order. The row reduction yields: Gröbner basis is a convenient tool to actually compute 2 x various mathematical objects in the commutative algebra. xye xf1 1100000 xe Example 5.9 Elimination of variables: The intersection of ef 0011000 1 2 = ye ideal IR⊂ and the subring S⊂ Rx[ 12, x ,..., xn ] is computable. f 000− 1010 4 y2 Let S= Rx[ tn+1,..., x]and let G be a Gröbner basis of I . Then f5 0100101−− y GGSt = ∩ is the Gröbner basis of S. If we compute the Gröbner 1 basis by means of lexicographic orders xx12> >>... xn , we 2 2 obtain the set of polynomials, We obtain f4 =−+ ye y and f5 =−+− xye y 1 and now
i we have the temporary Gröber basisG= { fff145,,} ; f2 and { fx11( n ), i= 1,..., i} , f3 can be excluded in the consideration, because they are i , “top-reducible” by f1 and easily recovered by the present G. { fx21( nn− , x) , i= 1,..., i2} ... Let us compose the third order Macaulay matrix: i = , xye { fnt−+11( x t +,..., x n − 1 , xi n) , 1,... int−−1} y ef 1100 ye22 ... 1 = 2 f5 −−101 1 y fi xx, ,..., x , i= 1,..., i . { n( 12 nn) } 1
Thus we can easily get GGSt = ∩ . The row reduction yields this:
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xye Stickelberger’s theorem y ef 110 0 ye22 In [2], an alternative of the numerical solution, instead 1 = 2 of Gröbner basis, is also used. This method is based on f6 011− 1 y 1 Stickelberger’s theorem. In general, for a zero-dimensional ideal I⊂= R kx[ 1,... xn ], RI/ is a k dimension; that is to say, the vector space is spanned by We obtain f= ye22 +− y 2 1. Now G= { fff,,} 6 146 the set of monomials and the result-vector of the space multiplication of finite We again compute the third order Macaulay matrix between two monomial is also represented by the linear combination of the monomial basis. Therefore, according to 22 ye the assertion of the theorem, the operation of a monomial to yf4 −11 02 = y the monomial basis is thought to be the operation of a matrix − f6 11 1 1 in k . And the eigenvalue of the matrix gives the numeral value of the corresponding monomial at VI( ) . The row reduction yields: 22 Consider I=( xyeyxex +,, + + y −⊂= 1) R[ xy ,] . As a Gröbner 22 ye basis of I is ( x+ ye, 2 y22 −− 1, e 1) , the monomial basis in RI/ is yf4 −11 02 = y given by{1,ye , e , y} , from which x is dropped. − f7 02 1 1 The transformation by y and e are given as follows.
=2 − 1 y We obtain fy7 21. Now G= { fff,,}. 147 2 ye ye We compute the fourth order Macaulay matrix: y = e ye 22 2 ye y y yf4 2 −11 0 0y y 1 = 2 − 2 ef7 1 0 1/2 0 e 1/2e 2 = 1 ye We do the row reduction in the fourth order Macaulay 1/2 matrix: 00011 22 ye 0 0 1/2 0 ye yf = . 4 −11 0 0 y2 0100e = 1 2 1/2000 f 0 1− 1/2 0 e y 2 8 1
22 1 e We obtain f=2 ye − . Now G= { ffff,,,}. However, 2 8 1478 ye ye − 2 2 = as ff78 yields fe= −1, and as f= − yf, G would be e 2 9 49 e e 22 { f179,, f f} =+{ x ye, 2 y −− 1, e 1}. The s-polynomials for G are y ye computed now: e 11ye y spoly f, f=+= x y3 e f + f = ( 17) 1 7 1 2 22 ye 3 , spolyf( 19, f) =+=+ x ye f1 yef 9 00101 0001 ye =22 −=−. = . spoly( f79,2 f) y e f7 f 9 1000e As those s-polynomials reduce to zero with respect to G, 0100y it is proved that G is a Gröbner basis. The transformation matrices M y (by y ) and M e (by e) are In the above example, we process the computation those in the right hand side in the both equations. It is easily carefully so that the unnecessary polynomials are removed checked that there are four eigenvectors, common both for immediately as soon as they appear; we also limit the M y and M e , which lie in RI/ : computation in the Macaulay matrices which would be as small as possible, although these matrices might be embedded 1 11 11 into a very large one. Indeed we have done the row reduction ye 1/ 2−− 1/ 2 1/ 2 1/ 2 numerically, not symbolically, in a very large, but sparse = ,, e 11−− 11 matrix. The algorithms F4 and F5 policies in a systematical way so that these algorithms are of y 1/ 2−− 1/ 2 1/ 2 1/ 2 by Faugère adopt these the most efficient methods to generate Gröbner bases.J Multidis Res Rev 2019 66 www.innovationinfo.org Now we have obtained the numeral values of e and y at mathematical foundation of such computations is the
VI( ) from y and y . In fact, the eigenvalues M y and M e also primary ideal decomposition. give us the value of y and e at VI( ). As for the value of x , we easily compute it from the polynomial relation by the ideal I. Primary ideal decomposition There are two special sorts of ideal: prime ideal and Algorithm of computing Krull dimension primary ideal, as we have seen in the previous section. As we have seen, the zero-dimensional ideal is detected One can comprehend these sorts of ideal with the analogy immediately after the computation of its Gröbner ideal. of elementary number theory. An integer is decomposed aa12 an However, it takes a little of algebra to compute the non-zero as n= pp12 pn by the prime factorization. The ideal of dimension of an ideal [40]. the integer n is n (the integer multiple of n ); and it is represented by np=aa12 ∩ p ∩∩ pan as the intersection Let I⊂ Rx,..., x be an ideal. Then the Krull 12 n • [ 1 n ] of the ideals generated by certain powers of primes. dimension of Rx[ 1,..., xn ] / I is given by an integer d
, i.e., dimKn(Rx[ 1 ,..., x] / I) = d, such that d is the maximal An ideal in the commutative algebra, likewise, could be
cardinality of a subset of variables uxx⊂ { 1,..., n} with decomposed as the intersection of primary ideals [3]: I∩= Ru[ ] (0) ( A maximal independent set of variables Ip= i with respect to I is a subset u of variables of maximal i One of the most elementary procedure for primary ideal decomposition is summarized as follows [41]. Let I be the • Let > be any monomial ordering on Rx[ 1,..., xn ]. And let cardinality, defined as this.) ideal in a Noetherian ring R , and rs, ∈ R are elements such I⊂ Rx[ 1,..., xn ] be an ideal. Then the Krull dimension of the quotient ideal is computed by means of the initial ideal that rI∉ , sI∉ and rs∈ I . ∞ n in< (I ) of I: 1. Find n such that Ir::= Ir. = n n dimKn(Rx[ 11 ,..., x] / I) dim Kn( Rx[ ,..., x] / in< ( I)) 2. Let IIr1 = + ( ) and I2 = Ir: ( ) .
Hence we only have to consider the initial ideal in< (I ) , 3. The ideal I1 and I2 are larger than I . The decomposition instead of the ideal I itself. process must be done for each of them. By choosing
some proper r , we can decompose I and I2 , hence I by • Let I=( m11,..., mkn) ⊂ Rx[ ,..., x]be an ideal with monomial 1 primary ideals. generators mi . Let denote {xx1,..., n} by X d( IRx, [ ]) in a recursive way Example 6.1 Consider J=++( x ey, y ex) in R= [ xye,,]with . Define d((0,) RX[ ]) = n, the lexicographic order xye>>. and • y is not in the radical J ; and ye⋅( 2 −=1.) eyxey( +) − yyex( +) ∈ J Thus we set r = y and dIRX( ,[ ]) = max{ dI( |x=01 , RX[ \ x ii]) for x such that x i | m }. i se=2 −1 Then d( IKx,[ ]) = dimK ( Rx[ ] / I) . • Let J1 ≡( J, y) =++( y ,, x ye y xe) . This is a primary (in fact, a I=( xy) ⊂ x, y 2 Example 5.14 Consider [ ] . prime) ideal, and Jy:= Jy : = ...(stabilized).
d( I,[ xy ,]) = max{ d(( 0,) [ x]) , d(( 0,) [ y])} , 2 • Let J2 ≡ J: y =−+ ( e 1, x ye ) : this is because of another representation J= y e2 −+1, x ye . The ideal J can be as II|xy=00= |0 = = ( ) . Since d((0,) [ xd]) = (( 0,) [ y]) = 1, ( ( ) ) 2 . decomposed again. d( I,[ xy ,]) = dimK [ xy ,] /( xy) = 1 Example 5.15 Consider I=( xy −⊂1,) [ x y]. Then we have to • Now take re= −1, se= +1. consider in< (I) = ( xy) and the conclusion is the same as the • J21≡( Je 2 , −=−+ 1) ( exeyexy 1, ) =−+( 1, ) . This ideal is previous example. primary (in fact, prime). Example 5.16 Consider I=+( x ey, y +⊂ ex) [ x,, y e] with ≡ −=+ − 2 • J22( Je 2 : 1) ( e 1, xy) and Je22:( −= 1) Je : ( − 1) = ... lexicographic order xye>>. The Gröbner basis is (stabilized). This ideal is primary (in fact, prime). 2 {y−+ ye, x ey} . Hencein< (I) = ( yx , ) . Since ue= { } is the set of variable of maximal cardinality with Now we have done the primary ideal decomposition:
JJ= 1∩∩ J 21 J 22 . ( yx,0) ∩=[ e] ( ) We should notice that the ideal J is the secular equation we have dimxye , , / I= 1. (We can arrive at the same K [ ] of the diatomic system, conclusion by means of the recursive function d( I ,[ xye , ,] ).) 01− xx Algorithm for the Decomposition of the ideal = e as eigenvalue solver −10 yy In the example of the hydrogen molecule in section 3, The primary ideal decomposition is the operation we have seen that the polynomial secular equation is made equivalent to the eigenstate computation by means of linear into the triangular form, in which the relation of variables algebra: the solutions of the secular equation are given by the affine algebraic sets of the computed ideals. We obtain the is clarified, almost to represent the roots themselves. The J Multidis Res Rev 2019 67 www.innovationinfo.org
eigenvalue e =1and e = −1 for which the eigenvectors are iii) If IK( u)[ X\ u] = QQ1 ∩∩ s is an irredundant ( xx, ) and ( xx,− ) . primary decomposition, then IK( u)[ x\ u]∩ K[ X ] =Q∩∩ KX[ ] ∩∩... Q KX[ ] is also an irredundant primary In fact, the above example is one of the simplest cases ( 1 ) ( s ) decomposition. which we can compute manually. The general algorithm for
Example 6.2 Consider I=++ x ey, y ex in [xye,,] with practical algorithms are developed by Gianni, Trager and lexicographic order xye>>. We know a Gröbner basis of ideal decomposition is first proposed by Eisenbud [42]; and Zacharias [43], by Shimoyama and Yokoyama [44], by Möller I is{y( e2 −+1,) x ey} . [45] and by Lazard . (As for the semi-algebraic set, one can do • Choose uy= { }as a maximal independent set of variables. similar decomposition, as is studied by Chen and Davenport 2 [46]). The comparison of algorithms is given in [10]. • I( y)[ x, e] =−+( e 1, x ey) . This ideal is a Gröbner basis in ( y)[ xe, ]. Algorithm of Gianni, Trager, and Zacharias 2 2 • Then h = lcm( lc(e−+1) ,lc( x ey)) = xe . In this section, we review the primary ideal decomposition J:= Ih : = Ih :22 =−+ ( e 1, x ey ) 2 • [xye,,] . And(Ih,) =++( xeyyexxe ,,) =( xy ,) . algorithm of Gianni, Trager, and Zacharias (GTZ algorithm) We have a decomposition I=( I:, h) ∩( Ih) . [18, 40]. In the algorithm which we have seen in the previous We decompose . As is zero-dimensional in ( y)[ xe,,] section, we have to search a “key polynomial” by trial and • J J we apply the algorithm in zero-dimensional case. error to decompose an ideal. In contrast, GTZ algorithm Since e2 −=1( ee + 11)( −) , the decomposition of J is given by J=( Je,( + 1))∩∩( Je ,( −=+− 1)) ( e 1, x y) ( e −+1, x y) . As the ideal J is in general position with respect to lexicographic order Definitionenables us to6.1 find such a key in a more rational way. xe> in ( y)[ xe, ], (Je,1( + )) and (Je,1( − )) are primary ideals. ⊂ • A maximal ideal IMn Kx[ 1,..., x] is called in general Notice that we are working in ( y)[ xe, ]. However, as the position with respect to the lexicographical ordering decomposition of J[ xye,,] inherits that of J( y)[ xe, ] , > ∈ with xx1 >>... n , if there exist g1,..., gnn Kx[ ] such that we have obtained the required decomposition.
I=++( x gx( ),..., x−− g( x), gx( )) . M 11n n1 n 1 n nn Triangulation of polynomial system • When an ideal is decomposed by primary ideal In case of the systems of polynomial equations decomposition IQ= 1 ∩∩ Qs , the prime ideals PQii= {pin( xx12 , ,... x) | i= 1... m } are called associated primes of I. Pi is a minimal associated prime of I if PP≠ for all ji≠ . solutions (i.e. to be zero-dimensional), several methods ij methods are proposed which[45,47] have which only decompose finitely many the • A zero-dimensional ideal I∈ Kx[ 1,..., xn ] is called in general position with respect to the monomial order > with with respect to the variables entries, xx> ), if all associated primes PP,..., are in general 1 n 1 s solutionll set into finitely many l subset of the triangular form ∩∩≠ { fx1( 1 ), fxx 1 ( 12 , ),..., fxxnn ( 12 , ,... xl )= 1,... L} position and Pi Kx[ nj] P Kx[ n] for ij≠ . This kind of algorithm is used in the application of (Zero-dimensional Decomposition) Let Theorem 6.1 algebraic geometry in molecular orbital theory as is I∈ Kx[ 1,..., xn ] be a zero-dimensional ideal. Let ( f) = I ∩ Kx[ ] , n demonstrated in section 3, instead of conventional linear ν1 ν s ff= 1 fs with ff≠ for ij≠ . Then the decomposition of ij algebra. ideal I is given by Let us review the triangular decomposition algorithm by s Möller [45]. The algorithm is based upon several lemmas. I= (, Ifvi ). i i=1 Lemma 6.1 (Lemma 2 in [45]) Let A be an ideal in a ring
R= [ xx1,..., n ] such that Krull dimension of the residue If I is in general position with respect to the monomial class ring RA/ is zero : dim( R/0 A) = . If is an ideal such that νi B order > with xx1 >>... n , then (If, ) are primary ideals for BA⊂ and if m∈ all i . m set VA( ) is the disjoint union: V( A) = V( AB: ) ∪ VB( ) with is sufficiently large, the affine algebraic Theorem 6.2 (Ideal decomposition in general case) Let VB( ) ={ y ∈ V( A) | ∀∈ b Bby :( ) = 0} Xx= ( 1,..., xn ) . Let I KX[ ] be an ideal, and let uX⊂ be a maximal independent set of variables for I . Then these V( AB:m ) = { y ∈ V( A) | ∃∈ b Bby :( ) ≠ 0} . statements hold. VB( ) of this theorem, given in [45], is i) The ideal IK( u)[\ X u ] generated by I in Ku( )[ xu\ ] is zero- slightly different from the conventional one. In this section (The definition of K with variables u by Ku( ) . dimensional. We denote the field of rational functions on Lemmaonly, we 6.2 use (Lemma this definition.) 3 in [45]) Let A is a ideal in a ring R such ii) Let {gg1,... s} ⊂ I be a Gröbner basis of IK( u)[ x\ u] . Let that Krull-dimension of RA/ equals to zero dimkR( / A) = 0 ∞ h= lcm( lc( g1 ) ,...,lc( gs ))∈ Ku[ ] . Then = . IK( u)[ x\: u]∩ K[ x] I h and let B be another ideal in R such that AB⊆=( g1,..., gs ) . dd+1 =dd ∩ If Ih::= Ih , then I( I : h ) (, Ih ). mm,1 ,..., ms ∈
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s The reduced Gröbner basis with respect to the V AB:m = V ( A + g ,..., g : gmi . ( ) ( ( 11ii− ) ) lexicographic order is given by i=1 xye>>
mi with V(( A+( g11 ,..., gii−−) : g) =∈===≠ { y VA( ) | g1( y) ... gi 1( y) 0 gi( y) }. 22 In addition, if A is a radical ( AA= ), the above relation holds p12=++ x y e, p = 2 y + 2 ye + e − 1, for all positive mm, ,..., m. 2 3 42 1 s p34=22 ye + y ++ e e , p =−− e e 2 di − f:= gx ,..., x xdji Lemma 6.3 (Lemma 5 iii) in [45]). Let i∑ ij ( 11n− ) n j=0 We apply the triangulation to A0= ( pppp 1234,,,) . For with nonzero polynomials g , ir=1,..., . If Fff:= { 1 ,..., r } is a ij variable x , it exists only in p . Thus p should be added in Gröbner basis with respect to a monomial order <, then 1 1 all components of the decomposition. We only decompose Gg= { 10,... gr 0} is a a Gröbner basis with respect to <. the smaller system A( ppp234,,) . For the variable y , 2 42 Lemma 6.4 (Lemma 7 in [45]). Let Gf:= { 1 ,..., fr }be a reduced pe3 =22 + , pee 4 =−−2 . By the removal of the redundant
Gröbner basis with respect to a monomial order <, where xn term, the simpler Gröbner basis for ( pp34, ) is, which is also is lexicographically in front of {xx11,.., n− } , and let a Gröbner basis of ( pp34,:) p 2.
di − = dji As for the decomposition =∩+2 m , fi:∑ gx ij ( 11 ,..., xn− ) x n V( A) VB( ) VA( :( e 1) ) j=0 the components are given as = + with nonzero polynomials gij , ir=1,..., , and B( pp34,:) p 2( p 2) lt( fr ) << lt( f1 ) . If g is a constant, then the ideal quotient 10 =(e2 +1, 2 y 2 + 2 ye +−= e 2 1) ( e 22 + 1, y + ye − 1) ( f21,..., ffr ) : has the Gröbner basis (with respect to <)
{gg20,..., r } and f12∉( ff,...r ) }. and { ff,..., } 22 Indeed, if 1 r generates a zero-dimensional ideal, Ae:( += 1) ( e − 2, 2 ye +) g is constant by Lemma 6 in [45]. 10 Indeed, the last statement implies the saturation: since, The algorithm is summarized as follows: for m =1,2,... , Let A(⊂ Rx[ ,..., x] ) be a zero-dimensional ideal with a 22m 1 n Ae: (+=− 1)( e 2, 2 ye +) reduced Gröbner basis { ff1,..., r } with respect to a monomial order <. Let B: = (f ,...,f ): f +(f ). Then V (A) is the union Thus we obtain the decomposition of ( ffff1234,,,) as the 2 r 1 1 22 2 m intersection of and . as V (A) = V (A:B V (B). (It is easily checked that for (e+1, y ++ ye 1, p1 ) (e−+2, 2 y ep , 1 ) ideals I, J, the relation V(I J) = V(I V(J) holds. Hence the Simple Example of The Molecular Orbital ) ∪ decomposition Method by Means of Algebraic Geometry of ideal: KIJ= ∩ .) The varieties∩ involved)∪ in the union have following properties.of the affine algebraic set is equivalent to that Let us execute molecular orbital computation by means of algebraic geometry. The example is a hexagonal molecule, For VB( ) (which is one of the components of the like benzene, where s-orbital is located at each atom and = decomposition), it holds that VB( ) V( f12)∩ V( f,..., fr ) and interacts only with the nearest neighbors. as is depicted in { ff2 ,..., r } is a reduced Gröbner basis. Figure 4. By lemma 4.4, V( AB: m ) is the disjoint union of the zero- The secular equation for a hexagonal molecule, like a zets, benzene, could be given by the simplest model:
mm12++ mr VAf( :,1) VA(( ( f1)) : f 2) ,..., VA(( ( ff12 , ,..., frr− 1)) : f) These sets are other components of the decomposition of VA( ) . A A m 5 4 VA(:( f1 )) = ∅ (because fA1 ∈ ): this item is removed. The Gröbner bases are computable for the ideals m m 2 ff)) : r Aff+ ( 12) : , ... , A+ (, ff12 , ..., rr−1 . These Gröbner bases are in Rx[ 11,..., xn− ] (where the dimension is exactly dropped by one). Therefore we go down into A A Rx[ 11,..., xn− ] 6 3 and in this ring we have the set of ideals which are to be processed again.
Iteratively we apply the algorithm to the set of ideals and drop the variable one by one; the process shall terminate A1 A2 Let us consider the problem Figure 4: The framework of a hexagonal molecule. The atoms are indexed after finite steps. as AA16,..., and they interact between nearest neighbors, as are indicated by 2 2 22 the bonds. f123=+ x yef,, =+ y xef =+− x y 1
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0TT 000 cc11 Table 2: Gröbner basis of the secular equation of the hexagonal molecule. TT0 000cc22 _[1]=e^4-5*e^2*T^2+4*T^4 0000TT cc33 _[2]=6*c6^2*e^2-6*c6^2*T^2-e^2+T^2 = ε 0000TTcc44 _[3]=2*c5*e^2*T-2*c5*T^3-c6*e^3+c6*e*T^2 000TT 0 cc55 _[4]=c5*e^3-c5*e*T^2-2*c6*e^2*T+2*c6*T^3 TT000 0cc66 _[5]=4*c5^2*T^2-4*c5*c6*e*T-4*c6^2*e^2 We place one orbital in each of the vertex of the hexagon; +8*c6^2*T^2+e^2-2*T^2 we assume the interaction between the nearest neighbors _[6]=4*c5^2*e-4*c5*c6*T+4*c6^2*e-e with the hopping parameter T; the wavefunction is given by _[7]=24*c5^2*c6^2*T-4*c5^2*T-24*c5*c6^3*e (cc16,..., ) to6 six atomic orbitals. (We assume +4*c5*c6*e+24*c6^4*T-10*c6^2*T+T the following model: ψ = Caii such that aa = δ ; and 6 ∑ i j ij, the= coefficients i=1 = H∑ aTaii+1 in the hexagonal closed graph with aa71.) _[8]=24*c5^4*T-16*c5^3*c6*e+16*c5^2*c6^2*T i=1 -10*c5^2*T+8*c5*c6^3*e+2*c5*c6*e-4*c6^2*T+T The corresponding energy functional is given by _[9]=c4*T-c5*e+c6*T
U = Tc21 ( c + c 3 ) + Tc 32 ( c + c 4 )+ Tc 43 ( c + c 5 ) _[10]=c4*e+12*c5^3*T-8*c5^2*c6*e+8*c5*c6^2*T
+Tc54 ( c + c 6 ) + T 65 ( c + c 1 ) + Tc 16 ( c + c 2 ) -4*c5*T+4*c6^3*e 222222 -ecccccc (1 + 23456+ + + + -1) _[11]=c3*T-c4*e+c5*T The ideal from the secular equation is represented as: _[12]=c3*e-4*c4^2*c5*e+12*c4^2*c6*T+4*c4*c5^2*T I=(-2*c1*e+2*c2*T+2*c6*T, -8*c4*c5*c6*e+12*c5^2*c6*T+8*c6^3*T-4*c6*T _[13]=c2*T-c3*e+c4*T 2*c1*T-2*c2*e+2*c3*T, _[14]=c2*e-c3*T+c5*T-c6*e 2*c2*T-2*c3*e+2*c4*T, _[15]=c1*T+c5*T-c6*e 2*c3*T-2*c4*e+2*c5*T, _[16]=c1*e-c2*T-c6*T 2*c4*T-2*c5*e+2*c6*T, _[17]=c1^2+c2^2+c3^2+c4^2+c5^2+c6^2-1 2*c1*T+2*c5*T-2*c6*e, Table 3: The Gröbner basis of the ideal If + ( ) . -c1^2-c2^2-c3^2-c4^2-c5^2-c6^2+1). _[1]=4*T^4-5*T^2*U^2+U^4 We assume that _[2]=e-U _[3]=12*c6^2*T^2-12*c6^2*U^2-e^2+3*e*U-2*T^2 I⊂ [ ccccccTeU123456,,,,,,,, ] _[4]=c5*T^2*U-c5*U^3+c6*e^2*T-2*c6*T^3+c6*T*U^2 with the lexicographic monomial order _[5]=2*c5*T^3-2*c5*T*U^2+c6*e^3-2*c6*e*T^2+c6*T^2*U
ccccccTeU123456>>>>>>>> _[6]=4*c5^2*U-4*c5*c6*T+4*c6^2*e-e The Gröbner basis is given in Table 2: _[7]=4*c5^2*T^2-4*c5*c6*e*T-4*c6^2*e*U+8*c6^2*T^2+e*U-2*T^2 _[8]=24*c5^2*c6^2*T-4*c5^2*T-24*c5*c6^3*e+4*c5*c6*e between the energy e and the hopping integral T. Then the +24*c6^4*T-10*c6^2*T+T The first entry of the list in Table 2 shows the relation polynomials including other variables (c6, c5,...., c1) appear in _[9]=24*c5^4*T-16*c5^3*c6*e+16*c5^2*c6^2*T- succession. This is the example of variable elimination. 10*c5^2*T+8*c5*c6^3*e +2*c5*c6*e-4*c6^2*T+T Let us evaluate the energy functional U. For this purpose, we make a new ideal If+ ( ) with a polynomial f which _[10]=c4*U+12*c5^3*T-8*c5^2*c6*e+8*c5*c6^2*T- 4*c5*T+4*c6^3*e equates the variable U functional at f = 0 : _[11]=c4*T-c5*e+c6*T and the definition of the energy _[12]=c3*U-4*c4^2*c5*e+12*c4^2*c6*T+4*c4*c5^2*T-8*c4*c5*c6*e f = Tc21 ( c + c 3 ) + Tc 32 ( c + c 43 ) c + Tc 43 ( c + c 5 ) +12*c5^2*c6*T+8*c6^3*T-4*c6*T +Tc54 ( c + c 6 ) + T 65 ( c + c 1 ) + Tc 16 ( c + c 2 ) 222222 _[13]=c3*T-c4*e+c5*T -eccccccU (123456 + + + + + -1)- _[14]=c2*U-c3*T+c5*T-c6*e The Gröbner basis of the ideal If+ ( ) is given in Table 3. The _[15]=c2*T-c3*e+c4*T U and the hopping integral, while the other variables are _[16]=c1*U-c2*T-c6*T sweptfirst polynomial into remaining gives polynomials.the relation between Consequently, the total it followsenergy _[17]=c1*T+c5*T-c6*e that, by means of symbolic computation, we have executed _[18]=c1^2+c2^2+c3^2+c4^2+c5^2+c6^2-1 the molecular orbital computation and have obtained the
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Table 4: Primary ideal decomposition of ideal If + ( ) . The ideal Table 5: Gröbner basis for the ideals pfi + ( ) . decomposes into ve primary ideals, each of which is represented by the generators. [1](p1+(f)) [1]:(p1) _[1]=U
_[1]=T _[2]=T
_[2]=e _[3]=e
_[3]=c1^2+c2^2+c3^2+c4^2+c5^2+c6^2-1 _[4]=c1^2+c2^2+c3^2+c4^2+c5^2+c6^2-1 [2]:(p2) _[1]=e-T [2](p2+(f)) _[2]=4*c5^2-4*c5*c6+4*c6^2-1 _[1]=T-U _[3]=c4-c5+c6 _[2]=e-T _[4]=c3+c6 _[3]=4*c5^2-4*c5*c6+4*c6^2-1 _[5]=c2+c5 _[4]=c4-c5+c6 _[6]=c1+c5-c6 _[5]=c3+c6 [3]:(p3) _[6]=c2+c5 _[1]=e+T _[7]=c1+c5-c6 _[2]=4*c5^2+4*c5*c6+4*c6^2-1 _[3]=c4+c5+c6 _[4]=c3-c6 [3](p3+(f)) _[5]=c2-c5 _[1]=T+U _[6]=c1+c5+c6 _[2]=e+T [4]:(p4) _[3]=4*c5^2+4*c5*c6+4*c6^2-1 _[1]=e+2*T _[4]=c4+c5+c6
_[2]=6*c6^2-1 _[5]=c3-c6
_[3]=c5+c6 _[6]=c2-c5
_[4]=c4-c6 _[7]=c1+c5+c6 _[5]=c3+c6 _[6]=c2-c6 [4](p4+(f)) _[7]=c1+c6 _[1]=2*T+U [5]:(p5) _[2]=e+2*T _[1]=e-2*T _[3]=6*c6^2-1 _[2]=6*c6^2-1 _[4]=c5+c6 _[3]=c5-c6 _[5]=c4-c6 _[4]=c4-c6 _[6]=c3+c6 _[5]=c3-c6 _[7]=c2-c6 _[6]=c2-c6 _[7]=c1-c6 _[8]=c1+c6 observable quantities: if we give T , we determine the total energy U and we have the relations for the variables [5](p5+(f)) of the wave-function and the eigenvalue e . As for the _[1]=2*T-U wave-function, there is a particular feature in the symbolic computation, which will be discussed later. _[2]=e-2*T _[3]=6*c6^2-1 Now let us see how the primary ideal decomposition _[4]=c5-c6 works. The decomposition for ideal If+ ( ) is given in Table 4. Each entry in the list is the primary ideal _[5]=c4-c6 = {pii | 1,...,5}, the intersection of which5 shall build the _[6]=c3-c6 polynomial ideal of secular equation Ip= ) . i (i= 1) _[7]=c2-c6 + Table 5 gives the Gröbner basis for the ideal ( pfi ( )) _[8]=c1-c6
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pf + ( ) Table 6: The dimension of the ideals i . For example, e = −3/2 is the eigenvalue with six fold degen- eracy and the corresponding eigenvectors are discrete (as dim (R/I) Ideal k zero-dimensional ideal) as follows:
pf1 + ( ) 5 (ccc123, ,) =±−±−±−( 1/ 2,1/ 2,0,) ( 1/ 2,0, 2,) ( 0,1/ 2, 2) pf + ( ) 2 2 Outlook for Related Topics pf + ( ) 3 2 We have seen how the computational algebraic geometry could be applied to the molecular orbital theory, in so far as pf4 + ( ) 1 the equations could be represented by polynomials. In this
pf5 + ( ) 1 section, we will see the related topics on the symbolic com- putation by means of polynomials. Polynomial optimization total energy U and the hopping parameter T . This is the Polynomial optimization is a numerical method which improvementThe first entry in the gives result the linearby the relationuse of primary between ideal the solves the optimization problem given by polynomial decomposition; in Table 3 of the Gröbner basis, we have equations and inequities [47,48-51]. obtained the relation between U and T , but the polynomial Example 8.1 Consider the cost function gxx( 12, ) = x 2and the is not decomposed yet. constraints 22 , =−≥, . f1=−−≥50 xx 12 f210 xx 12 f312=−≥ xx0 The dimension of the ideals is given in Table 6. (We refer The problem is given by: to the Krull dimension of the quotient ring by an ideal by the phrase “dimension of the ideal” according to the custom of Maximize the cost multivariate polynomial function gx( ) computational algebraic geometry.) with the constraint of multivariate polynomial function fx≥ As for pf1 + ( ) i ( ) 0. = = = and cc22++=1 UTe 16 In other words, the cost function g is to be maximized in + freedom for (cc16,..,, the) . As affine for pf algebraic( ) and set pf5 is( determined) , the only one by f1 , f2 and f3 , as in Figure 5. indeterminate is the hopping ; integralthus there T , arewhich five contributedegrees of The key idea is to replace the monomials xxij with one to the dimension of the ideal; (cc16,..., ) is determined up the semi-algebraic set defined by 12 to sign. As for pf2 + ( ) and pf( ) , the hopping integral T is variables yij . The variables should satisfy relations such as yy= y still an indeterminate and contributes one to the dimension; pq,, rs p++ qr , s and y00 =1 moment matrix in the following style: (cc16,..., ) is not determined explicitly, unless c5 or c6 would be explicitly given in the quadratic equation in the list (which is . Let us define the first-order 1 yy10 01 one-dimensional geometrical object). In this circumstance, = the dimension of the ideal rises by one. The increase of My1() y10 y 20 y 11 pf+ pf+ the dimension in the cases of 2 ( ) and 3 ( ) is, by the yyy01 11 02 familiar phrase of physics, due to the degeneracy in the eigenvalue. For such cases in the numerical computation, By means of the formalism of the polynomial optimization, the solvers automatically return the ortho-normalized the problem is restated as follows. basis vectors. The numerical library determines them by an arbitrary way. However, by summing up the degenerated eigenstates with the equal weights, one can compute the 3 observable quantum mechanical quantity which is free from this sort of ambiguity. In contrast, the symbolic computation 2 processes the equation up to the “minimal” polynomials of the variables of the wave-functions and it does not anymore. The wave-functions are, however, non-observable 1 quantities; one can eliminate them from the polynomial equations by symbolic computation so that one could obtain 0 only observable quantities, such as energy. In fact, it is not always true that the degeneracy of eigenvalues should lead to -1 non-zero-dimensional character of eigenspace: for example, consider the following polynomial as the energy functional: -2 f=−+− ccc ccc +− ccc +2 12( 3) 21( 3) 31( ) -3 444444 222 -3 -2 -1 0 1 2 3 +(cccccceccc122331 + +) −( 123 ++−1) Figure 5: The curves 5-xy22 - = 0 , yx= , 10−=xy are depicted. The y-coordinate of the upper-right intersection point between the diagonal line The local energy minima are represented by discrete and the hyperbola gives the solution of the optimization problem presented points (not continuous) both in real and in complex number. in the main article.
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The relaxation of order k is stated in this way: Maximize the linear cost function y01 , With the constraint that the moment matrix is = α My1 ( ) pky* min ∑ ααfy
such that Myk ( ) 0 semi-positiveWith the constraint definite, of multivariate polynomial functions: ˆ ˆ ˆ f1 =−−≥50 yy20 02 , fy=−≥10, fyy=−≥0 and M Gy 0 , = 2 11 3 10 01 kd− i ( i ) i 1,2,... In the above, the constraint of the moment matrix comes It is guaranteed that, as the degree k increases, the * optima pk converges to the global optimum p. 22 from the≤ semi-positive-definiteness of the quadratic form: Example 8.2 Let us compute these matrices. 0 ∑∑cxii cx ii ∑ cc ijij y i=11i= ij , The second-order moment matrix is given by
1 yyyyy10 01 20 11 02 matrix as . M M 0 yyyyyy10 20 11 30 21 12 Hereafter we denote the semi-positive definiteness of a yyyyyy01 11 02 21 12 03 The optimization problem is solved by the standard way My2 ()= yyyyyy20 30 21 40 31 22 yyyyyy the dual problem also. In general, it is not guaranteed that 11 21 12 31 22 13 yyyyyy of semi-positive-definite programming. We can formulate 02 12 03 22 13 04 correct answer; indeed the formulation should be given The higher-order moment matrices are computed the use of the first-moment matrix would suffice to give the likewise. The localizing matrices are computed for the polynomials in the matricesabove formulation, of infinite there dimension, is no whichconstraint include for the f1 , f2 and f3 in the example problem. conditionmoments yy ofpq,, higher rs= y porders++ qr , s. We up only to the expect infinity. that the In addition,moment 22 matrix would be reduced into the one with lower and For f1=−−5 xx 12, suitable rank at the optimum. Actually, the above procedure 55−−yy20 02 yyy 10 −− 30 12 5 yyy 01 −− 21 03 is an approximation and the accuracy is improved by the use = −− −− −− of larger matrices. Mfyyyy1() 1 510 30 12 5 yyy 20 40 225 yyy 11 31 13 5yyy10−− 21 02 55 yyy 11 −− 31 13 yyy 02 −− 22 04 α α1 αn Let yy= ()α n , where yα represents x= xx ... . Let α∈ n 1 n y αα= denote the degree of ( α ) as ∑ i . We try to optimize the i=1 For f2=1 − xx 12, problem in the limited range of ()yα n , such that α ≤ 2r . α∈ 1−y11 yyyy10 −− 21 01 12 For y= ( yyαα, ,..., y α)(such that α N ≤ 2r ), the r-th order 11 N moment matrix My is constructed as follows: M() fy= y −− y y y − y y − y r ( ) 1 2 10 21 20 20 31 11 22 yy01− 12 yy 11 −− 22 yy 02 13 Myr ( )[1,1] = 1 , M( yi) ,1 = y, r [ ] αi For f3= xx 12 − , , My( )[1, i] = yα r i yy10− 01 yyyy 20 −− 11 11 02 M y ij, = y , r ( )[ ] ααij+ =−−− Mfyyyyyyy1() 2 20 11 30 21 21 12 where the index i, j are taken in the range such that αi ≤ r . yy11−−− 02 yyyy 21 12 12 03 px( ) Let be the polynomial constraint : Observe that these localizing matrices include the entries β of the second order moment matrix and do not contain px()= ∑ cβ x beta
For px( ) concisely with the use of My2 ( ) , M11( fy), M12( fy) , M13( fy) . superfluous ones. We can optimize the given problem As is demonstrated in [2], once the molecular orbital M( py), the i, j localizing= c y matrix is defined by: s [ ] ∑ βααij++ β theory has been built over the polynomial system on the β ground of algebraic geometry and commutative algebra, it where the index are taken in the range such that ij, enables us to joint the equation of conventional quantum α ≤ s i mechanics and the extra constraints into a set of polynomial In general,. This matrix a global should polynomial be semi-positive optimization definite, problem too. is equations. We solve the problem through the collaboration given by of numerical and symbolical methods. The computational = = α p* miny Fx ( )∑ αα f x described by the set of equations. However, it is somewhat scheme is basically to determine the affine algebraic set such that Gx( ) ≥ 0 , i =1,2,.... inconvenient that we should use only equations for the i general optimization problem. It seems that polynomial − Assume that the degree of gxi ( ) to be 21di or 2di . In optimization could get over such limitations because it could correspondence to the above global polynomial problem, deal with the constraints by inequities in the semi-algebraic
J Multidis Res Rev 2019 73 www.innovationinfo.org set. The mathematical ground is “real algebraic geometry” (ax,2)∈{ } ×( −∞ , ∞) , which contains a various range of interests. (ax,)∈( 2, ∞) ×( −∞ , ∞) . Quantifier elimination In each cylinder, we should check the zeros and the sign When we deal with equations and inequities of 2 condition of f( a,1 x) =++ x ax in ax− plane. The solutions polynomials, we often ask ourselves about the existence of of f( ax, ) are listed as follows. the solutions. For example, under what condition would a =−−2 − a ∈( −∞,2 − ) : the solutions are x1 ( aa4) /2 and =−+2 − . (QE) is the computational process to answer this sort of x2 ( aa4) /2 questions:quadratic equation when the havequestion roots? is given The “quantifierby ∃∈x.0( xelimination”2 + bx + c =) a = −2 : the solution is x1 =1. ∃ ) and returns the b2 −≥40 ac as the answer. In fact, there are a ∈−( 2, 2) : no solution. , the computer eliminates the quantifier ( a = 2 : the solution is x1 = −1. simplifiedproposed byform several mathematicians, such as by Fourier, by a ∈−( 2, ∞) : the solutions are x=−− aa2 −4 /2 and Motzkin,general theories and by forTarski executing [52,53], such or Presburger sort of simplifications, arithmetic. 1 ( ) =−+2 − . But those algorithms are not practical. Afterward, more x2 ( aa4) /2 practical algorithms by Collins and by others have come into (In the actual computation, we do not try to obtain such use, called “Cylindrical Algebraic Decomposition” (CAD). The analytic solutions in uplifting step. Instead, we place one theoretical ground of the standard algorithm in Algebraic sampling point, say as , in each subdivision, and we inspect Cylindrical Decomposition are given in [46,54-58]. There the sign condition and the solution of univariate polynomial are several computer packages in which this algorithm is fax( s , ) . ) implemented, such as QEPCAD [59], Mathematica [60], Maple [61], and Reduce [62]. The cylinders are again divided into cells according to the sign condition of f( ax, ) . For example, the cylinder Since CAD algorithm is apparently related to the theme (−∞,2 −) ×( −∞ , ∞) of this article, let us review the computational procedure in this section. Let us consider the problem through a simpler (−∞,2 −) ×( −∞ is ,x divided1 ) ; by five cells:
(−∞,2 −) ×{x }; 2 1 example∃a.( xof quantifier + ax += 10 elimination:) . (−∞,2 −) ×( xx , ) ; We can simplify the above prenex form into the one 12 2 −≥. a 40 (−∞,2 −) ×{x2} ; The cylindrical algebraic decomposition analyses the . without quantifier, such as (−∞,2 −) ×( x2 , ∞) critical point of f( a, x) = x2 + ax += 10 in two-dimensional real ax− plane. The critical points are the solutions of For other cylinders, we construct the cells likewise, 2 ∂f( ax, ) f( a, x) = x + ax += 10 and =2xa + . They are also the ∂x as is illustrated in Figure 6. These cells in a-x plane are solution of x2 + ax +=10, 20xa+= and x(20 xa+=) . Thus we obtain the matrix equation: 11ax 2 0 4 = 20ax 0 3 02a 1 0 2 The determinant of the matrix in the left-hand side f ax, is the discriminant of ( ) . If it is zero, the matrix 1 equation would permit non-zero vector solution. Hence we project the polynomial in one ring into a polynomial 0 in another ring of lower dimension (Projection step). We now analyse the projected polynomial (the discriminant) -1 in order to inquire after the existence of the root. As the discriminant is 4 − a2 -2 (−∞−−−, 2) ,{ 2} ,( 2, 2) ,{ 2} ,( 2, ∞) so that each of them gives the distinct sign of the discriminant., we can divide the a-axis into five cells: -3 Let us build the set of cylinders in a-x plane which pass -4 through each cells in a-axis. They are given by -4 -3 -2 -1 0 1 2 3 4 (ax,)∈( −∞ ,2 −) ×( −∞ , ∞) , Figure 6: The cylindrical algebraic decomposition for x2 ++ ax 1. The vertical and horizontal axes represent the variables x and a respectively. (ax,2,) ∈{ −} ×( −∞ ∞) , The zone is decomposed into cells by the solid lines and curves; each cell is distinguished from others by the sign conditions of two polynomials, (ax,)∈( − 2, 2) ×( −∞ , ∞) , x2 ++ ax 1 and a2 − 4 .
J Multidis Res Rev 2019 74 www.innovationinfo.org distinguished by the sign conditions of f( a,1 x) =++ x ax i j 2 = − and dis(fa )= 4 − . Then we seek the cells which would of {xAi | e 1,.,,,0} and {xB | j= d − 1,...,0} so that the productin which of thethe rowsmatrix are madeand the from vector the array ( xxde +of, coefficients de +−1 ,..., x ,1) of the roots of x2 + ax +=10, and by joining the cells which should give those set of polynomials. The discriminant of satisfy the condition of the first question abouta ≤− the2 existenceor a ≥ 2 . fx( ) is a special case as the resultant of fx( ) and df( x) . dx satisfyIn generalthe requirement, multivariate we find case the of answer: CAD, the algorithm executes the multi-step projection (from n-variables to one- to the molecular orbital theory when we give the polynomial We expect that quantifier elimination would be applicable variable ) and uplifting (from an axis to the whole space). goes well, we would obtain the polynomial representation of There are examples of QE with the taste of molecular therepresentation solution of tothe the optimization problem. If problem; the quantifier we could elimination render orbital theory. Let e,,( xy) be the energy and the wavefunction the zone and the boundary in the parameter space which of the simple diatomic system, such that shall satisfy our requirement; we might “reason” for the question of quantum mechanics by means of the rigorous 01− xx foundation of logic. However, at present, the computer and = e the algorithm are still powerless; indeed the complexity of −10 yy the algorithm, in the worst case, is double exponential in
22 the number of variables . (This is because of the enormous Example 8.3 ∃ee: <∧+ 0 xey =∧+ 0 yex =∧ 0 xy + −=∧+ 1 0 xy = 0.. number of combinations of polynomials in the process The prenex form inquires if the negative e (the energy) of variable-elimination.) Therefore we must expect some would exists in the diatomic molecule of asymmetric breakthrough both in algorithm and in computer architecture xy+=0 concludes that the formula is practical purpose. configuration . The quantifier elimination in order that we could apply quantifier elimination for the False . Polynomial optimization and wave-geometry In the above examples, we implicitly assume that all of the Example 8.4 required ingredients are polynomials. They are generated ∃ee: <∧+ 0 xey =∧+ 0 yex =∧ 0 xy22 + −=∧−= 10 xy 0 by the method of quantum chemistry: by the use of localized atomic basis, the integrodifferential equation is converted The prenex form inquires if the negative e (the energy) into the secular equation of analytic functions, and the latter would exists in the diatomic molecule of symmetric is furthermore approximated by polynomials. On the other xy−=0 hand, the fundamental equations of quantum mechanics always involve differential operators. Indeed the symbolic configuration . The quantifier elimination computation would be applicable to the algebra generated 22+ −= ∧ − = concludesx y that10 the formula xy is 0 simplified as by variables and differentials (G-algebra); the Gröbner basis could be computed by extending the Buchberger’s algorithm form contains several polynomials {fjn( xx12 , ,..., x) | j= 1,..., m } (Mora’s algorithm [63]). However, G-algebra is rather a This is the process of quantifier elimination. If the prenex purely mathematical topic and it seems to be powerless in , we have to compute the intersection of fi and f j for ij≠ solving numerical problems. as well as the critical point of each fi . The intersection is computed by projection by means of variable elimination, For this issue, the aforementioned polynomial optimiza- tions would give us some hint. The key idea is to replace the the tool of which is “resultant”. Assume that xn are the uni-variate polynomials of x monomials in the equation with the variables of one-degree, n and the latter variables are determined by the framework of the polynomials of {xx,.., − } . In this assumption, the 11n, in which the coefficients are resultant for two polynomials The algorithm of polynomial optimization is based on the =dd +−1 ++ semi-positive definite programming [64]. A ax01 ax ad measure theory. The variables of degree one, corresponding α αα11 and to the monomials X= xx12 ⋅ , represent the integrals in
ee−1 the moment problem, B= bx01 + bx ++ be = α µ is the determinant of a square matrix of dimension d+e, yα ∫ Xd by means of some probability measure µ
aa01 ad 00 dµ =1. We construct the entry of the moment matrix in the 00a aa ∫ defined 01asdd− , which satisfies following form 00aa01 ad αβ bb b 0 yαβ, = XXdµ 01 e ∫ 00bb01e− Instead of directly determining the measure, however, 00bb b 01 e the algorithm computes the moments yαβ, by utilizing the
J Multidis Res Rev 2019 75 www.innovationinfo.org constraints among them so that the objective function would 22 −aMuu[ ] + bMxx[ ] +− ab( Mux[ ] + Mxu[ ]) be maximized. = −+−aMuu22[ ] bMxx[ ] abh M[−− uu] h /2 a it is plausible that, if we inspect the problem from the = (ab,0)(≥ ) analogyThis foundationof quantum of mechanics, the algorithm the has measure a great dsignificance:µ might be −h/2 M[ zz] b replaced by ||φ ( x) 2 dx by a certain “wavefunction” φ , and the moments will be given by the expectation value in the replaced by that of the matrix. sense of quantum mechanics. As the expectation values of The semi-positive definiteness of the quadratic form is the products of coordinate operators and the differential operators can be computed, we can extend the idea of the demands these relations: moment matrix. For the univarite case, the typical integrals Semi-positive definiteness of the moment matrix ≥ are given by M[ xx] 0 ij M[−≥ uu] 0 yij = (φφ xx ) h2 or M[ xx] M[−−≥ uu] 0 4 j ∂ det( )≥ 0 yx= φφi . ij: ∂x MxxMxMx[ ]−≥[ ] [ ] 0 Let us consider the simplest problem: the harmonic (The diagonal entries of the matrix are positive; the oscillator. values of determinant of 2 by 2 minors, taken along the diagonal, are positive; and the determinant of the matrix is The total energy of the harmonic oscillator is given by the positive.) sum of squares of the kinetic-momentum operator p and 22 the coordinate operator xH: = p + x, with the relation: With this condition, the optimum of E= M[ −+ uu] M[ xx] 1 is obtained at M[ xx] =−= M[ uu] h and Mx[ ] = 0 . The solution [p,x] =−−1h 2 might be obtained by several methods. The authors of this In order to avoid the argument in the complex number, article (Kikuchi and Kikuchi) had tried several algorithms to p we make use of u= hd/ dx , instead of . solve them in the sequence of works: the solution by means of QE in [65]; by means of particle swarm optimization in [66]; by means of directly determining the measure µ itself i j kl i j kl WeMxu define the xu extended = ∫ ( xu momentφφ)( xu matrix) dx as in [67]. This sort of integral is rearranged by the linear combination of those terms can “quantize” the polynomial equations by providing the d mn mn Maybe the significance of this algorithm is that one variables {xi} with the differential operators h ; the search M xu= ∫φφ( xu) dx. dx of the roots of the polynomial Wx( ) is replaced by the ground- d 2 The the moment matrix (indexed by 1,xu , ) is given as: =−+2 state computation of the quantum Hamiltonian H h2 Wx( ) dx 10Mx[ ] ; in the limit h → 0 , the “probability distribution” of the Mx[ ] Mxx[ ] − h/2 . set VW. It is not so audacious to say that we discover the 0−−h /2 M[ uu] ( ) quantumseminal idea system of “wave shall geometry”, coincide withwhich the is affinethe counterpart algebraic (In the above, M[−=− uu] M[ uu] due to the linearity of the in mathematics to the “wave mechanics” in physics. In the constant.) In order to compute that matrix, we have used use middle of twentieth century, in fact, Mimura et al. proposed the relations: the idea of “wave geometry” [68]: their fundamental idea d is to assume the line elements 2 in differential geometry M[ u][10] = ∫ hφφ dx = ds dx as the operator in quantum mechanics, to which, therefore, d dd d2 the differential operator h is coupled. In contrast to that M u u= hφφφ h dx =−=− h2 φ dx M uu ds [ ][ ] ∫∫ 2 [ ] dx dx dx theory, our idea of “wave geometry” is based upon algebraic geometry, and we expect that our idea would be applicable dd M[ u][ x] = hφφ( x) dx=−⋅ φ h x φ dx =− M[ ux] to quantitative – numeric or symbolic – computation by ∫∫dx dx means of powerful techniques developed for quantitative d = φφ−−h x dx =− M[ xu] − h. simulation of quantum dynamics and also by means of the ∫ dx modern theory of algebra. In order to see the necessity of h /2, let us consider the quadratic form: Available Packages of Symbolic Computation There are several packages of symbolic computation ∫ (au+ bx)φφ( au +≥ bx) dx( 0) by which we can compute Gröbner bases or primary ideal This quadratic form is represented by decomposition.
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• CoCoA : Computer algebra system [69]. case of symbolic-numeric computation of molecular • GAP: Software for computational discrete algebra. The chief In this article, at first, we have demonstrated a model aim of the package is to execute the computation in the group orbital theory with the taste of algebraic geometry. Then theory, but it could process polynomials [70]. As for the we have expounded the related mathematical concepts application of GAP software in material physics, see the work in commutative algebra and algebraic geometry with of Kikuchi [71], and the article by Kikuchi and Kikuchi [72]. simple examples. In addition, we have introduced several mathematical and computational methods, which would • Macaulay 2: Computer algebra system [73,74]. be connected to this post-contemporary theory of quantum • Mathematica: Technical computing system for almost all chemistry. The two main principles of the theory are to represent the involved equations by polynomials and to process the polynomials by computer algebra. Then one • Maple: Symbolic computation software for general fields of science and industry [60]. can analyze the polynomial equations and unravel the purpose. The primary ideal decomposition is implemented at the package “Regular Chains Package” [61]. dependence between the variables. In the traditional molecular orbital theory, the principal mathematical tool is • Maxima: Symbolic computation software for general the linear algebra. Indeed, it is a subset of the commutative purpose [75]. algebra. For instance, the diagonalization of the matrix and • Reduce: Computer algebra system [62]. the orthonormality of eigenstates would be comprehensible in a wider context: the primary ideal decomposition. And SINGULAR: Computer algebra system [76]. It contains • the latter has a close relation to the other fundamental ideas various libraries to solve the problems in algebraic in algebraic geometry: the resolution of singularity and geometry. It also contains the extension package “Plural” the normalization of the variety. Besides, the polynomial for non-commutative algebra. One can learn how to approximation (inevitable in our theory) should ideally be compute by Singular from introductory textbooks [40,77]. embedded into the “completion” of commutative rings; we do • packages. sake of facile computations in the limited resources. Without As for quantifier elimination, also there are available this approximation with the finite number of symbols for the • doubt, there are a lot of other concepts in commutative elimination by means of Partial Cylindrical Algebraic algebra and algebraic geometry which would be embodied DecompositionQEPCAD: A free [59]. software, which does quantifier in the application of computational quantum mechanics. You should Ask, and it will be given to you... • Reduce Or, • Mathematica • Maple petite, et dabitur vobis; quaerite, et invenietis; pulsate, et aperietur vobis. • There is a platform system which bundles several free software packages in mathematical science. It will be a felicity for us, for the authors of this article, if you enjoy every bite of the “quantum chemistry with the • SageMath: Open-Source Mathematical Software System. view toward algebraic geometry” and if this article stirs From this platform, one can utilize a lot of software your curiosity; we heartily greet your participation in the packages both of numerical and symbolic computations research of this new theme. (in which Gap, Maxima, and Singular are included) [78]. • There are a lot of research centers of symbolic Acknowledgment computations. A great deal of computer algebra systems The authors are grateful to all readers who have spent are the products of the studies over long years in several the precious time to accompany with the authors through the seemingly “arid” topics toward the end of the article. the latest researches of INRIA in France: universities. One can find software implementations of References • INRIA: l’institut national de recherche dédié aux sciences du numérique. The research programs, at the interplay 1. Szabo A, Ostlund NS (2012) Modern quantum chemistry: introduction to of algebra, geometry and computer science, are ongoing advanced electronic structure theory. Courier Corporation. now [79]. 2. Kikuchi A (2013) computation by symbolic-numeric computation. QScience Connect. An approach to first principles electronic structure Summary 3. Eisenbud D (2013) Commutative Algebra: with a view toward algebraic Dear readers, our voyage has now ended up the planned geometry. Springer Science & Business Media. course. How do you think about the connection between 4. Reid M, Miles R (1995) Undergraduate commutative algebra. Cambridge quantum mechanics and algebraic geometry? We have found University Press. it even in the familiar region of quantum chemistry. Algebraic 5. Perrin D (2007) Algebraic geometry: an introduction. Springer Science geometry is not a language of another sphere; it is a tool to solve & Business Media. actual problems with quantitative precision. Is it interesting for 6. Cox DA, Little J, O’shea D (2006) Using algebraic geometry. Springer you? Or have you judged coldly that it is a plaything? Science & Business Media.
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7. Cox D, Little J, OShea D (2013) Ideals, varieties, and algorithms: an 33. Bardet M, Faugere JC, Salvy B (2015) On the complexity of the f5 Gröbner introduction to computational algebraic geometry and commutative basis algorithm. Journal of Symbolic Computation 70: 49-70. algebra. Springer Science & Business Media. 34. Faugere JC (1999) A new ecient algorithm for computing Gröbner bases 8. Becker T, Weispfenning V (1993) Gröbner bases. Gröbner Bases. (f4). Journal of pure and applied algebra 139: 61-88. Springer. 35. Faugere JC (2002) A new ecient algorithm for computing Gröbner bases 9. Ene V, Herzog J (2011) Gröbner Bases in Commutative Algebra. American without reduction to zero (f 5). Proceedings of the 2002 international Mathematical Society. symposium on Symbolic and algebraic computation. 10. Decker W, Jong T, Greuel GM, Pster G (1999) The normalization: a new 36. Faugere JC, Gianni P, Lazard D, Mora T (1993) algorithm, implementation and comparisons. Computational Methods of zero-dimensional Gröbner bases by change of ordering. Journal of for Representations of Groups and Algebras, Springer 173: 177-185. Symbolic Computation 16: 329-344. Efficient computation 11. Decker W, Lossen C (2006) Computing in algebraic geometry. Algorithms 37. Giovini A, Mora T, Niesi G, Robbiano L, Traverso C (1991) one sugar cube, and computation in mathematics. please” or selection strategies in the buchberger algorithm. Proceedings of the ISSAC’91. 12. Bochnak J, Coste M, Roy MF (2013) Real algebraic geometry. Springer Science & Business Media. 38. Brickenstein M (2010) Slimgb: Gröbner bases with slim polynomials. Revista Matematica Complutense 23: 453-466. 13. Lasserre JB (2015) An introduction to polynomial and semi-algebraic optimization,volume 52. Cambridge University Press. 39. Traverso C (1988) Gröbner trace algorithms. International Symposium on Symbolic and Algebraic Computation, Springer. 14. Crespo T, Hajto Z, Ruiz JJM (2007) Introduction to differential Galois theory. Wydawnictwo PK Cracow. 40. Decker W, Pster G (2013) geometry. Cambridge University Press. 15. Acosta-Humanez P, Blazquez-Sanz D (2006) Non-integrability of some A first course in computational algebraic hamiltonians with rational potentials. 41. Swanson I. Primary ideal decomposition. 16. Acosta-Humanez PB, Venegas-Gomez H (2018) Liouvillian solutions of 42. Eisenbud D, Huneke C, Vasconcelos W (1992) Direct methods for Schrödinger” equation with polynomial potentials using Gröbner basis. primary decomposition. Inventiones mathematicae 110: 207-235. 17. Lazard D (1983) Gröbner bases, gaussian elimination and resolution 43. Gianni P, Trager B, Zacharias G (1988) Gröbner basis and primary of systems of algebraic equations. European Conference on Computer decomposition of polynomial ideals. Journal of Symbolic Computation Algebra, Springer. 6:149-167. 18. Gianni P, Trager B, Zacharias G (1988) Gröbner bases and primary 44. Shimoyama T, Yokoyama K (1996) Localization and primary decomposition of polynomial ideals. Journal of Symbolic Computation decomposition of polynomial ideals. Journal of Symbolic Computation 6: 149-167. 22: 247-277. 19. Montes A, Wibmer M (2010) Gröbner bases for polynomial systems with 45. Moller HM (1993) On decomposing systems of polynomial equations parameters. Journal of Symbolic Computation 45: 1391-1425. Communication and Computing 4: 217-230. 20. Buchberger B (1965) with finitely many solutions. Applicable Algebra in Engineering, class ring of a zerodimensional polynomial ideal. 46. Chen C, James, John PM, Maza MM, Xia B, et al. (2013) Triangular An algorithm for finding a basis for the residue decomposition of semi-algebraic systems. Journal of Symbolic 21. Buchberger B (1965) Ein algorithmus zum aunden der basiselemente Computation 49: 3-26. des restklassenringesnach einem nulldimensionalen polynomideal. 47. Lazard D (1992) Solving zero-dimensional algebraic systems. Journal of 22. Gebauer R, Moller HM (1988) On an installation of buchberger’s symbolic computation 13: 117-131. algorithm. Journal of Symbolic Computation 6: 275-286. 48. Lasserre JB (2001) Global optimization with polynomials and the 23. Ene V, Herzog J (2012) Gröbner bases in Commutative Algebra. American problem of moments. SIAM Journal on optimization 11: 796-817. Mathematical Society. 49. Blekherman G, Pablo AP, Rekha RT (2012) 24. Hermann G (1998) The question of nitely many steps in polynomial and convex algebraic geometry. SIAM. ideal theory. ACM SIGSAM Bulletin 32: 8-30. Semidenfinite optimization 50. Gloptipoly 3: moments, 25. Dube TW (1990) The structure of polynomial ideals and Gröbner bases. optimization and semidenite programming. Optimization Methods & SIAM Journal on Computing 19: 750-773. SoftwareHenrion D,24: Jean-Bernard 761-779. L, Lofberg J (2009) 26. Wiesinger-Widi M (2011) Gröbner bases and generalized sylvester 51. Laurent M (2009) Sums of squares, moment matrices and optimization matrices. ACM Comm.Computer Algebra 45: 137-138. over polynomials. Emerging applications of algebraic geometry, Springer. 27. Ernst W Mayr, Albert R Meyer (1982) The complexity of the word problems for commutative semigroups and polynomial ideals. Advances 52. Tarski A (1931) Sur les ensembles denissables de nombres reels. in mathematics 46: 305-329. Fundamenta mathematicae 17: 210-239. 28. Huynh DT (1986) A superexponential lower bound for Gröbner bases 53. Tarski A (1998) A decision method for elementary algebra and geometry. and church-rosser commutative thue systems. Information and Control Quantier elimination and cylindrical algebraic decomposition, Springer. 68: 196-206. 54. Arnon DS, Collins GE, McCallum S (1984) Cylindrical algebraic 29. Giusti M (1984) Some selectivity problems in polynomial ideal theory. decomposition I: The basic algorithm. SIAM Journal on Computing 13: International Symposium on Symbolic and Algebraic Manipulation, 865-877. Springer. 55. Arnon DS, George EC, McCallum S (1984) Algebraic decomposition II: 30. Michael Moller H, Mora F (1984) Upper and lower bounds for the degree an adjacency algorithm for the plane. SIAM Journal on Computing 13: of Gröbner bases. International Symposium on Symbolic and Algebraic 878-889. Manipulation, Springer. 56. McCallum S (1993) Solving polynomial strict inequalities using 31. Macaulay FS (1902) Some formulae in elimination. Proceedings of the cylindrical algebraic decomposition. The Computer Journal 36: 432-438. London Mathematical Society 1: 3-27. 57. Chen C, Maza MM (2016) Quantier elimination by cylindrical algebraic 32. Giusti M (1985) A note on the complexity of constructing standard bases. decomposition based on regular chains. Journal of Symbolic Computation In European Conference on Computer Algebra, Springer. 75:74-93.
J Multidis Res Rev 2019 78 www.innovationinfo.org
58. Chaouki TA, Dorato P, Liska R, Steinberg S, Yang W (1995) Applications 70. The Gap Group (2019) Gap - groups, algorithms, programming - a system of quantier elimination theory to control theory. for computational discrete algebra. 59. Hong H, Christopher WB. Qepcad version B. 71. Kikuchi A (2018) Computer Algebra and Materials Physics: A Practical Guidebook to Group Theoretical Computations in Materials Science. 60. Wolfram Inc. Symbolic computation software mathematica. Springer. 61. Maplesoft. Maple. 72. Kikuchi I, Kikuchi A (2017) Lie algebra in quantum physics by means of 62. Hearn AC, REDUCE Developers. Reduce computer algebra system. computer algebra. arXiv preprint. 63. Mora T (1994) An introduction to commutative and noncommutative 73. Grayson D, Stillman M, Eisenbud S (1992) Macaulay 2. gröbner bases. Theoretical Computer Science 134: 131-173. 74. Eisenbud D, Grayson DR, Stillman M, Sturmfels B (2001) Computations 64. Lasserre JB (2006) Convergent sdp-relaxations in polynomial in algebraic geometry with Macaulay 2. Springer Science & Business optimization with sparsity. SIAM Journal on Optimization 17: 822-843. Media. 65. Kikuchi I, Kikuchi A (2018) Polynomial optimization and quantier 75. Project Maxima. Maxima, a computer algebra system. elimination in quantum mechanics. OSF Preprints. 76. Decker W, Greuel GM, Pster G, Schoenemann H Computer algebra system 66. Kikuchi I, Kikuchi A (2018) A formulation of quantum particle swarm singular. optimization. OSF Preprints. 77. Greuel GM, Pster G (2012) A Singular introduction to commutative 67. Kikuchi I, Kikuchi A (2018) Quantum mechanical aspect in bayesian algebra. Springer Science & Business Media. optimization. OSF Preprints. 78. The Sage Foundation. Sagemath - free open-source mathematics 68. Mimura Y, Takeno H (1962) Wave geometry. Sci Rept Res Inst Theoret software system. Phys, Hiroshima Univ. 79. INRIA. L’institut national de recherche d’edie aux sciences du numerique. 69. Abbott J, Bigatti AM, Robbiano L (2006) CoCoA: a system for doing Computations in Commutative Algebra.
Citation: Kikuchi A, Kikuchi I (2019) Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post Contemporary Quantum Chemistry. J Multidis Res Rev Vol: 1, Issu: 2 (47-79).
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