18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #13 10/22/2013 13.1 Affine Varieties
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Prime Ideals in Polynomial Rings in Several Indeterminates
PROCEEDINGS OF THE AMIERICAN MIATHEMIIATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 67 -74 S 0002-9939(97)03663-0 PRIME IDEALS IN POLYNOMIAL RINGS IN SEVERAL INDETERMINATES MIGUEL FERRERO (Communicated by Ken Goodearl) ABsTrRAcr. If P is a prime ideal of a polynomial ring K[xJ, where K is a field, then P is determined by an irreducible polynomial in K[x]. The purpose of this paper is to show that any prime ideal of a polynomial ring in n-indeterminates over a not necessarily commutative ring R is determined by its intersection with R plus n polynomials. INTRODUCTION Let K be a field and K[x] the polynomial ring over K in an indeterminate x. If P is a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[xl,...,Xn]. Now, K [x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials fi, f..n such that P is minimal over (fi, ..., fn), the ideal generated by { fl, ..., f}n ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows that any prime ideal of K[xl, X.., Xn] is determined by n polynomials. -
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]. -
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. -
Algebra 557: Weeks 3 and 4
Algebra 557: Weeks 3 and 4 1 Expansion and Contraction of Ideals, Primary ideals. Suppose f : A B is a ring homomorphism and I A,J B are ideals. Then A can be thought→ of as a subring of B. We denote by⊂ Ie ( expansion⊂ of I) the ideal IB = f( I) B of B, and by J c ( contraction of J) the ideal J A = f − 1 ( J) A. The following are easy to verify: ∩ ⊂ I Iec, ⊂ J ce J , ⊂ Iece = Ie , J cec = J c. Since a subring of an integral domain is also an integral domain, and for any prime ideal p B, A/pc can be seen as a subring of B/p we have that ⊂ Theorem 1. The contraction of a prime ideal is a prime ideal. Remark 2. The expansion of a prime ideal need not be prime. For example, con- sider the extension Z Z[ √ 1 ] . Then, the expansion of the prime ideal (5) is − not prime in Z[ √ 1 ] , since 5 factors as (2 + i)(2 i) in Z[ √ 1 ] . − − − Definition 3. An ideal P A is called primary if its satisfies the property that for all x, y A, xy P , x P⊂, implies that yn P, for some n 0. ∈ ∈ ∈ ∈ ≥ Remark 4. An ideal P A is primary if and only if all zero divisors of the ring A/P are nilpotent. Since⊂ thsi propert is stable under passing to sub-rings we have as before that the contraction of a primary ideal remains primary. Moreover, it is immediate that Theorem 5. -
6. Localization
52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i. -
Prime Ideals in B-Algebras 1 Introduction
International Journal of Algebra, Vol. 11, 2017, no. 7, 301 - 309 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7838 Prime Ideals in B-Algebras Elsi Fitria1, Sri Gemawati, and Kartini Department of Mathematics University of Riau, Pekanbaru 28293, Indonesia 1Corresponding author Copyright c 2017 Elsi Fitria, Sri Gemawati, and Kartini. This article is distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce the definition of ideal in B-algebra and some relate properties. Also, we introduce the definition of prime ideal in B-algebra and we obtain some of its properties. Keywords: B-algebras, B-subalgebras, ideal, prime ideal 1 Introduction In 1996, Y. Imai and K. Iseki introduced a new algebraic structure called BCK-algebra. In the same year, K. Iseki introduced the new idea be called BCI-algebra, which is generalization from BCK-algebra. In 2002, J. Neggers and H. S. Kim [9] constructed a new algebraic structure, they took some properties from BCI and BCK-algebra be called B-algebra. A non-empty set X with a binary operation ∗ and a constant 0 satisfying some axioms will construct an algebraic structure be called B-algebra. The concepts of B-algebra have been disscussed, e.g., a note on normal subalgebras in B-algebras by A. Walendziak in 2005, Direct Product of B-algebras by Lingcong and Endam in 2016, and Lagrange's Theorem for B-algebras by JS. Bantug in 2017. -
Quadratic Form - Wikipedia, the Free Encyclopedia
Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form Quadratic form From Wikipedia, the free encyclopedia In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form). Contents 1 Introduction 2 History 3 Real quadratic forms 4 Definitions 4.1 Quadratic spaces 4.2 Further definitions 5 Equivalence of forms 6 Geometric meaning 7 Integral quadratic forms 7.1 Historical use 7.2 Universal quadratic forms 8 See also 9 Notes 10 References 11 External links Introduction Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: where a,…,f are the coefficients.[1] Note that quadratic functions, such as ax2 + bx + c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the 1 of 8 27/03/2013 12:41 Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. -
Solving Quadratic Equations in Many Variables
Snapshots of modern mathematics № 12/2017 from Oberwolfach Solving quadratic equations in many variables Jean-Pierre Tignol 1 Fields are number systems in which every linear equa- tion has a solution, such as the set of all rational numbers Q or the set of all real numbers R. All fields have the same properties in relation with systems of linear equations, but quadratic equations behave dif- ferently from field to field. Is there a field in which every quadratic equation in five variables has a solu- tion, but some quadratic equation in four variables has no solution? The answer is in this snapshot. 1 Fields No problem is more fundamental to algebra than solving polynomial equations. Indeed, the name algebra itself derives from a ninth century treatise [1] where the author explains how to solve quadratic equations like x2 +10x = 39. It was clear from the start that some equations with integral coefficients like x2 = 2 may not have any integral solution. Even allowing positive and negative numbers with infinite decimal expansion as solutions (these numbers are called real numbers), one cannot solve x2 = −1 because the square of every real number is positive or zero. However, it is only a small step from the real numbers to the solution 1 Jean-Pierre Tignol acknowledges support from the Fonds de la Recherche Scientifique (FNRS) under grant n◦ J.0014.15. He is grateful to Karim Johannes Becher, Andrew Dolphin, and David Leep for comments on a preliminary version of this text. 1 of every polynomial equation: one gives the solution of the equation x2 = −1 (or x2 + 1 = 0) the name i and defines complex numbers as expressions of the form a + bi, where a and b are real numbers. -
Degree Bounds for Gröbner Bases in Algebras of Solvable Type
Degree Bounds for Gröbner Bases in Algebras of Solvable Type Matthias Aschenbrenner University of California, Los Angeles (joint with Anton Leykin) Algebras of Solvable Type • . introduced by Kandri-Rody & Weispfenning (1990); • . form a class of associative algebras over fields which generalize • commutative polynomial rings; • Weyl algebras; • universal enveloping algebras of f. d. Lie algebras; • . are sometimes also called polynomial rings of solvable type or PBW-algebras (Poincaré-Birkhoff-Witt). Weyl Algebras Systems of linear PDE with polynomial coefficients can be represented by left ideals in the Weyl algebra An(C) = Chx1,..., xn, ∂1, . , ∂ni, the C-algebra generated by the xi , ∂j subject to the relations: xj xi = xi xj , ∂j ∂i = ∂i ∂j and ( xi ∂j if i 6= j ∂j xi = xi ∂j + 1 if i = j. The Weyl algebra acts naturally on C[x1,..., xn]: ∂f (∂i , f ) 7→ , (xi , f ) 7→ xi f . ∂xi Universal Enveloping Algebras Let g be a Lie algebra over a field K . The universal enveloping algebra U(g) of g is the K -algebra obtained by imposing the relations g ⊗ h − h ⊗ g = [g, h]g on the tensor algebra of the K -linear space g. Poincaré-Birkhoff-Witt Theorem: the canonical morphism g → U(g) is injective, and g generates the K -algebra U(g). If g corresponds to a Lie group G, then U(g) can be identified with the algebra of left-invariant differential operators on G. Affine Algebras and Monomials N Let R be a K -algebra, and x = (x1,..., xN ) ∈ R . Write α α1 αN N x := x1 ··· xN for a multi-index α = (α1, . -
Hyperbolicity and Stable Polynomials in Combinatorics and Probability
Hyperbolicity and stable polynomials in combinatorics and probability Robin Pemantle 1,2 ABSTRACT: These lectures survey the theory of hyperbolic and stable polynomials, from their origins in the theory of linear PDE’s to their present uses in combinatorics and probability theory. Keywords: amoeba, cone, dual cone, G˚arding-hyperbolicity, generating function, half-plane prop- erty, homogeneous polynomial, Laguerre–P´olya class, multiplier sequence, multi-affine, multivariate stability, negative dependence, negative association, Newton’s inequalities, Rayleigh property, real roots, semi-continuity, stochastic covering, stochastic domination, total positivity. Subject classification: Primary: 26C10, 62H20; secondary: 30C15, 05A15. 1Supported in part by National Science Foundation grant # DMS 0905937 2University of Pennsylvania, Department of Mathematics, 209 S. 33rd Street, Philadelphia, PA 19104 USA, pe- [email protected] Contents 1 Introduction 1 2 Origins, definitions and properties 4 2.1 Relation to the propagation of wave-like equations . 4 2.2 Homogeneous hyperbolic polynomials . 7 2.3 Cones of hyperbolicity for homogeneous polynomials . 10 3 Semi-continuity and Morse deformations 14 3.1 Localization . 14 3.2 Amoeba boundaries . 15 3.3 Morse deformations . 17 3.4 Asymptotics of Taylor coefficients . 19 4 Stability theory in one variable 23 4.1 Stability over general regions . 23 4.2 Real roots and Newton’s inequalities . 26 4.3 The Laguerre–P´olya class . 31 5 Multivariate stability 34 5.1 Equivalences . 36 5.2 Operations preserving stability . 38 5.3 More closure properties . 41 6 Negative dependence 41 6.1 A brief history of negative dependence . 41 6.2 Search for a theory . 43 i 6.3 The grail is found: application of stability theory to joint laws of binary random variables . -
Resultant and Discriminant of Polynomials
RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results are well-known 19th century mathematics, but I have not inves- tigated the history, and no references are given. 1. Resultant n m Definition 1.1. Let f(x) = anx + ··· + a0 and g(x) = bmx + ··· + b0 be two polynomials of degrees (at most) n and m, respectively, with coefficients in an arbitrary field F . Their resultant R(f; g) = Rn;m(f; g) is the element of F given by the determinant of the (m + n) × (m + n) Sylvester matrix Syl(f; g) = Syln;m(f; g) given by 0an an−1 an−2 ::: 0 0 0 1 B 0 an an−1 ::: 0 0 0 C B . C B . C B . C B C B 0 0 0 : : : a1 a0 0 C B C B 0 0 0 : : : a2 a1 a0C B C (1.1) Bbm bm−1 bm−2 ::: 0 0 0 C B C B 0 bm bm−1 ::: 0 0 0 C B . C B . C B C @ 0 0 0 : : : b1 b0 0 A 0 0 0 : : : b2 b1 b0 where the m first rows contain the coefficients an; an−1; : : : ; a0 of f shifted 0; 1; : : : ; m − 1 steps and padded with zeros, and the n last rows contain the coefficients bm; bm−1; : : : ; b0 of g shifted 0; 1; : : : ; n−1 steps and padded with zeros. In other words, the entry at (i; j) equals an+i−j if 1 ≤ i ≤ m and bi−j if m + 1 ≤ i ≤ m + n, with ai = 0 if i > n or i < 0 and bi = 0 if i > m or i < 0. -
Algebraic Combinatorics and Resultant Methods for Polynomial System Solving Anna Karasoulou
Algebraic combinatorics and resultant methods for polynomial system solving Anna Karasoulou To cite this version: Anna Karasoulou. Algebraic combinatorics and resultant methods for polynomial system solving. Computer Science [cs]. National and Kapodistrian University of Athens, Greece, 2017. English. tel-01671507 HAL Id: tel-01671507 https://hal.inria.fr/tel-01671507 Submitted on 5 Jan 2018 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. ΕΘΝΙΚΟ ΚΑΙ ΚΑΠΟΔΙΣΤΡΙΑΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΘΗΝΩΝ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΚΑΙ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ Μελέτη και επίλυση πολυωνυμικών συστημάτων με χρήση αλγεβρικών και συνδυαστικών μεθόδων Άννα Ν. Καρασούλου ΑΘΗΝΑ Μάιος 2017 NATIONAL AND KAPODISTRIAN UNIVERSITY OF ATHENS SCHOOL OF SCIENCES DEPARTMENT OF INFORMATICS AND TELECOMMUNICATIONS PROGRAM OF POSTGRADUATE STUDIES PhD THESIS Algebraic combinatorics and resultant methods for polynomial system solving Anna N. Karasoulou ATHENS May 2017 ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ Μελέτη και επίλυση πολυωνυμικών συστημάτων με