Solving Systems of Polynomial Equations Bernd Sturmfels
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Ladder Operators for Lam\'E Spheroconal Harmonic Polynomials
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 074, 16 pages Ladder Operators for Lam´eSpheroconal Harmonic Polynomials? Ricardo MENDEZ-FRAGOSO´ yz and Eugenio LEY-KOO z y Facultad de Ciencias, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: [email protected] URL: http://sistemas.fciencias.unam.mx/rich/ z Instituto de F´ısica, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: eleykoo@fisica.unam.mx Received July 31, 2012, in final form October 09, 2012; Published online October 17, 2012 http://dx.doi.org/10.3842/SIGMA.2012.074 Abstract. Three sets of ladder operators in spheroconal coordinates and their respective actions on Lam´espheroconal harmonic polynomials are presented in this article. The poly- nomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lam´epolynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers n1 and n2 counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components L^x, L^y, L^z of the angular momentum connect pairs of the four species of poly- nomials of a chosen kind and angular momentum. The third set of operators, the cartesian componentsp ^x,p ^y,p ^z of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities. -
Field Theory Pete L. Clark
Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9. -
10 7 General Vector Spaces Proposition & Definition 7.14
10 7 General vector spaces Proposition & Definition 7.14. Monomial basis of n(R) P Letn N 0. The particular polynomialsm 0,m 1,...,m n n(R) defined by ∈ ∈P n 1 n m0(x) = 1,m 1(x) = x, . ,m n 1(x) =x − ,m n(x) =x for allx R − ∈ are called monomials. The family =(m 0,m 1,...,m n) forms a basis of n(R) B P and is called the monomial basis. Hence dim( n(R)) =n+1. P Corollary 7.15. The method of equating the coefficients Letp andq be two real polynomials with degreen N, which means ∈ n n p(x) =a nx +...+a 1x+a 0 andq(x) =b nx +...+b 1x+b 0 for some coefficientsa n, . , a1, a0, bn, . , b1, b0 R. ∈ If we have the equalityp=q, which means n n anx +...+a 1x+a 0 =b nx +...+b 1x+b 0, (7.3) for allx R, then we can concludea n =b n, . , a1 =b 1 anda 0 =b 0. ∈ VL18 ↓ Remark: Since dim( n(R)) =n+1 and we have the inclusions P 0(R) 1(R) 2(R) (R) (R), P ⊂P ⊂P ⊂···⊂P ⊂F we conclude that dim( (R)) and dim( (R)) cannot befinite natural numbers. Sym- P F bolically, we write dim( (R)) = in such a case. P ∞ 7.4 Coordinates with respect to a basis 11 7.4 Coordinates with respect to a basis 7.4.1 Basis implies coordinates Again, we deal with the caseF=R andF=C simultaneously. Therefore, letV be an F-vector space with the two operations+ and . -
Homogeneous Polynomial Solutions to Constant Coefficient Pde’S
HOMOGENEOUS POLYNOMIAL SOLUTIONS TO CONSTANT COEFFICIENT PDE'S Bruce Reznick University of Illinois 1409 W. Green St. Urbana, IL 61801 [email protected] October 21, 1994 1. Introduction Suppose p is a homogeneous polynomial over a field K. Let p(D) be the differen- @ tial operator defined by replacing each occurence of xj in p by , defined formally @xj 2 in case K is not a subset of C. (The classical example is p(x1; · · · ; xn) = j xj , for which p(D) is the Laplacian r2.) In this paper we solve the equation p(DP)q = 0 for homogeneous polynomials q over K, under the restriction that K be an algebraically closed field of characteristic 0. (This restriction is mild, considering that the \natu- ral" field is C.) Our Main Theorem and its proof are the natural extensions of work by Sylvester, Clifford, Rosanes, Gundelfinger, Cartan, Maass and Helgason. The novelties in the presentation are the generalization of the base field and the removal of some restrictions on p. The proofs require only undergraduate mathematics and Hilbert's Nullstellensatz. A fringe benefit of the proof of the Main Theorem is an Expansion Theorem for homogeneous polynomials over R and C. This theorem is 2 2 2 trivial for linear forms, goes back at least to Gauss for p = x1 + x2 + x3, and has also been studied by Maass and Helgason. Main Theorem (See Theorem 4.1). Let K be an algebraically closed field of mj characteristic 0. Suppose p 2 K[x1; · · · ; xn] is homogeneous and p = j pj for distinct non-constant irreducible factors pj 2 K[x1; · · · ; xn]. -
Selecting a Monomial Basis for Sums of Squares Programming Over a Quotient Ring
Selecting a Monomial Basis for Sums of Squares Programming over a Quotient Ring Frank Permenter1 and Pablo A. Parrilo2 Abstract— In this paper we describe a method for choosing for λi(x) and s(x), this feasibility problem is equivalent to a “good” monomial basis for a sums of squares (SOS) program an SDP. This follows because the sum of squares constraint formulated over a quotient ring. It is known that the monomial is equivalent to a semidefinite constraint on the coefficients basis need only include standard monomials with respect to a Groebner basis. We show that in many cases it is possible of s(x), and the equality constraint is equivalent to linear to use a reduced subset of standard monomials by combining equations on the coefficients of s(x) and λi(x). Groebner basis techniques with the well-known Newton poly- Naturally, the complexity of this sums of squares program tope reduction. This reduced subset of standard monomials grows with the complexity of the underlying variety, as yields a smaller semidefinite program for obtaining a certificate does the size of the corresponding SDP. It is therefore of non-negativity of a polynomial on an algebraic variety. natural to explore how algebraic structure can be exploited I. INTRODUCTION to simplify this sums of squares program. Consider the Many practical engineering problems require demonstrat- following reformulation [10], which is feasible if and only ing non-negativity of a multivariate polynomial over an if (2) is feasible: algebraic variety, i.e. over the solution set of polynomial Find s(x) equations. -
Good Reduction of Puiseux Series and Applications Adrien Poteaux, Marc Rybowicz
Good reduction of Puiseux series and applications Adrien Poteaux, Marc Rybowicz To cite this version: Adrien Poteaux, Marc Rybowicz. Good reduction of Puiseux series and applications. Journal of Symbolic Computation, Elsevier, 2012, 47 (1), pp.32 - 63. 10.1016/j.jsc.2011.08.008. hal-00825850 HAL Id: hal-00825850 https://hal.archives-ouvertes.fr/hal-00825850 Submitted on 24 May 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Good Reduction of Puiseux Series and Applications Adrien Poteaux, Marc Rybowicz XLIM - UMR 6172 Universit´ede Limoges/CNRS Department of Mathematics and Informatics 123 Avenue Albert Thomas 87060 Limoges Cedex - France Abstract We have designed a new symbolic-numeric strategy to compute efficiently and accurately floating point Puiseux series defined by a bivariate polynomial over an algebraic number field. In essence, computations modulo a well chosen prime number p are used to obtain the exact information needed to guide floating point computations. In this paper, we detail the symbolic part of our algorithm: First of all, we study modular reduction of Puiseux series and give a good reduction criterion to ensure that the information required by the numerical part is preserved. -
Algebra Polynomial(Lecture-I)
Algebra Polynomial(Lecture-I) Dr. Amal Kumar Adak Department of Mathematics, G.D.College, Begusarai March 27, 2020 Algebra Dr. Amal Kumar Adak Definition Function: Any mathematical expression involving a quantity (say x) which can take different values, is called a function of that quantity. The quantity (x) is called the variable and the function of it is denoted by f (x) ; F (x) etc. Example:f (x) = x2 + 5x + 6 + x6 + x3 + 4x5. Algebra Dr. Amal Kumar Adak Polynomial Definition Polynomial: A polynomial in x over a real or complex field is an expression of the form n n−1 n−3 p (x) = a0x + a1x + a2x + ::: + an,where a0; a1;a2; :::; an are constants (free from x) may be real or complex and n is a positive integer. Example: (i)5x4 + 4x3 + 6x + 1,is a polynomial where a0 = 5; a1 = 4; a2 = 0; a3 = 6; a4 = 1a0 = 4; a1 = 4; 2 1 3 3 2 1 (ii)x + x + x + 2 is not a polynomial, since 3 ; 3 are not integers. (iii)x4 + x3 cos (x) + x2 + x + 1 is not a polynomial for the presence of the term cos (x) Algebra Dr. Amal Kumar Adak Definition Zero Polynomial: A polynomial in which all the coefficients a0; a1; a2; :::; an are zero is called a zero polynomial. Definition Complete and Incomplete polynomials: A polynomial with non-zero coefficients is called a complete polynomial. Other-wise it is incomplete. Example of a complete polynomial:x5 + 2x4 + 3x3 + 7x2 + 9x + 1. Example of an incomplete polynomial: x5 + 3x3 + 7x2 + 9x + 1. -
F[X] Be an Irreducible Cubic X 3 + Ax 2 + Bx + Cw
Math 404 Assignment 3. Due Friday, May 3, 2013. Cubic equations. Let f(x) F [x] be an irreducible cubic x3 + ax2 + bx + c with roots ∈ x1, x2, x3, and splitting field K/F . Since each element of the Galois group permutes the roots, G(K/F ) is a subgroup of S3, the group of permutations of the three roots, and [K : F ] = G(K/F ) divides (S3) = 6. Since [F (x1): F ] = 3, we see that [K : F ] equals ◦ ◦ 3 or 6, and G(K/F ) is either A3 or S3. If G(K/F ) = S3, then K has a unique subfield of dimension 2, namely, KA3 . We have seen that the determinant J of the Jacobian matrix of the partial derivatives of the system a = (x1 + x2 + x3) − b = x1x2 + x2x3 + x3x1 c = (x1x2x3) − equals (x1 x2)(x2 x3)(x1 x3). − − − Formula. J 2 = a2b2 4a3c 4b3 27c2 + 18abc F . − − − ∈ An odd permutation of the roots takes J =(x1 x2)(x2 x3)(x1 x3) to J and an even permutation of the roots takes J to J. − − − − 1. Let f(x) F [x] be an irreducible cubic polynomial. ∈ (a). Show that, if J is an element of K, then the Galois group G(L/K) is the alternating group A3. Solution. If J F , then every element of G(K/F ) fixes J, and G(K/F ) must be A3, ∈ (b). Show that, if J is not an element of F , then the splitting field K of f(x) F [x] has ∈ Galois group G(K/F ) isomorphic to S3. -
January 10, 2010 CHAPTER SIX IRREDUCIBILITY and FACTORIZATION §1. BASIC DIVISIBILITY THEORY the Set of Polynomials Over a Field
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION §1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics. A polynomials is irreducible iff it cannot be factored as a product of polynomials of strictly lower degree. Otherwise, the polynomial is reducible. Every linear polynomial is irreducible, and, when F = C, these are the only ones. When F = R, then the only other irreducibles are quadratics with negative discriminants. However, when F = Q, there are irreducible polynomials of arbitrary degree. As for the integers, we have a division algorithm, which in this case takes the form that, if f(x) and g(x) are two polynomials, then there is a quotient q(x) and a remainder r(x) whose degree is less than that of g(x) for which f(x) = q(x)g(x) + r(x) . The greatest common divisor of two polynomials f(x) and g(x) is a polynomial of maximum degree that divides both f(x) and g(x). It is determined up to multiplication by a constant, and every common divisor divides the greatest common divisor. These correspond to similar results for the integers and can be established in the same way. One can determine a greatest common divisor by the Euclidean algorithm, and by going through the equations in the algorithm backward arrive at the result that there are polynomials u(x) and v(x) for which gcd (f(x), g(x)) = u(x)f(x) + v(x)g(x) . -
Non-Openness and Non-Equidimensionality in Algebraic Quotients
Pacific Journal of Mathematics NON-OPENNESS AND NON-EQUIDIMENSIONALITY IN ALGEBRAIC QUOTIENTS JOHN DUSTIN DONALD Vol. 41, No. 2 December 1972 PACIFIC JOURNAL OF MATHEMATICS Vol. 41, No. 2, 1972 NON-OPENNESS AND NON-EQUIDIMENSIONALΠΎ IN ALGEBRAIC QUOTIENTS JOHN D. DONALD Suppose given an equivalence relation R on an algebraic variety V and the associated fibering of V by a family of subvarieties. This paper treats the question of the existence of a quotient structure for this situation when the fibering is non-equidimensional. For this purpose a general definition of quotient variety for algebraic equivalence relations is used which contains no topological requirements. The results are of two types. In §1 it is shown that certain maps into nonsingular varieties are quotient maps for the induced equivalence relation whenever the union of the excessive orbits has codimension ^ 2. This theorem yields many examples of non-equidimensional quotients. Section 2 contains a converse showing that no excessive orbit containing a normal hypersurface can be fitted into a quotient. This theorem depends on a stronger and less conceptual field- theoretic result which fails without the normality hypothesis. Section 3 contains a counterexample. These results are a first attempt at answering the question: when can a good algebraic structure be assigned to a non-equidimensional fibering of VΊ Analogously, one might ask whether there is a good sense in which an equidimensional family of subvarieties of V can degenerate into subvarieties of different dimension. It is a standard result that no fibers of a morphism have dimension less than that of the generic fiber, so that we must concern ourselves only with degeneration into higher-dimensional subvarieties. -
Section 1.2 – Mathematical Models: a Catalog of Essential Functions
Section 1-2 © Sandra Nite Math 131 Lecture Notes Section 1.2 – Mathematical Models: A Catalog of Essential Functions A mathematical model is a mathematical description of a real-world situation. Often the model is a function rule or equation of some type. Modeling Process Real-world Formulate Test/Check problem Real-world Mathematical predictions model Interpret Mathematical Solve conclusions Linear Models Characteristics: • The graph is a line. • In the form y = f(x) = mx + b, m is the slope of the line, and b is the y-intercept. • The rate of change (slope) is constant. • When the independent variable ( x) in a table of values is sequential (same differences), the dependent variable has successive differences that are the same. • The linear parent function is f(x) = x, with D = ℜ = (-∞, ∞) and R = ℜ = (-∞, ∞). • The direct variation function is a linear function with b = 0 (goes through the origin). • In a direct variation function, it can be said that f(x) varies directly with x, or f(x) is directly proportional to x. Example: See pp. 26-28 of the text. 1 Section 1-2 © Sandra Nite Polynomial Functions = n + n−1 +⋅⋅⋅+ 2 + + A function P is called a polynomial if P(x) an x an−1 x a2 x a1 x a0 where n is a nonnegative integer and a0 , a1 , a2 ,..., an are constants called the coefficients of the polynomial. If the leading coefficient an ≠ 0, then the degree of the polynomial is n. Characteristics: • The domain is the set of all real numbers D = (-∞, ∞). • If the degree is odd, the range R = (-∞, ∞). -
Computations in Algebraic Geometry with Macaulay 2
Computations in algebraic geometry with Macaulay 2 Editors: D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels Preface Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re- cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith- mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv- ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi- mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all.