Symmetries and Polynomials
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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 -
Introduction to Complex Analysis Michael Taylor
Introduction to Complex Analysis Michael Taylor 1 2 Contents Chapter 1. Basic calculus in the complex domain 0. Complex numbers, power series, and exponentials 1. Holomorphic functions, derivatives, and path integrals 2. Holomorphic functions defined by power series 3. Exponential and trigonometric functions: Euler's formula 4. Square roots, logs, and other inverse functions I. π2 is irrational Chapter 2. Going deeper { the Cauchy integral theorem and consequences 5. The Cauchy integral theorem and the Cauchy integral formula 6. The maximum principle, Liouville's theorem, and the fundamental theorem of al- gebra 7. Harmonic functions on planar regions 8. Morera's theorem, the Schwarz reflection principle, and Goursat's theorem 9. Infinite products 10. Uniqueness and analytic continuation 11. Singularities 12. Laurent series C. Green's theorem F. The fundamental theorem of algebra (elementary proof) L. Absolutely convergent series Chapter 3. Fourier analysis and complex function theory 13. Fourier series and the Poisson integral 14. Fourier transforms 15. Laplace transforms and Mellin transforms H. Inner product spaces N. The matrix exponential G. The Weierstrass and Runge approximation theorems Chapter 4. Residue calculus, the argument principle, and two very special functions 16. Residue calculus 17. The argument principle 18. The Gamma function 19. The Riemann zeta function and the prime number theorem J. Euler's constant S. Hadamard's factorization theorem 3 Chapter 5. Conformal maps and geometrical aspects of complex function the- ory 20. Conformal maps 21. Normal families 22. The Riemann sphere (and other Riemann surfaces) 23. The Riemann mapping theorem 24. Boundary behavior of conformal maps 25. Covering maps 26. -
On Algebraic Functions
On Algebraic Functions N.D. Bagis Stenimahou 5 Edessa Pella 58200, Greece [email protected] Abstract In this article we consider functions with Moebius-periodic rational coefficients. These functions under some conditions take algebraic values and can be recovered by theta functions and the Dedekind eta function. Special cases are the elliptic singular moduli, the Rogers- Ramanujan continued fraction, Eisenstein series and functions asso- ciated with Jacobi symbol coefficients. Keywords: Theta functions; Algebraic functions; Special functions; Peri- odicity; 1 Known Results on Algebraic Functions The elliptic singular moduli kr is the solution x of the equation 1 1 2 2F1 , ;1;1 x 2 2 − = √r (1) 1 1 2 2F1 2 , 2 ; 1; x where 1 2 π/2 ∞ 1 1 2 2 n 2n 2 2 dφ 2F1 , ; 1; x = 2 x = K(x)= (2) 2 2 (n!) π π 2 n=0 Z0 1 x2 sin (φ) X − q The 5th degree modular equation which connects k25r and kr is (see [13]): 5/3 1/3 krk25r + kr′ k25′ r +2 (krk25rkr′ k25′ r) = 1 (3) arXiv:1305.1591v3 [math.GM] 27 Mar 2014 The problem of solving (3) and find k25r reduces to that solving the depressed equation after named by Hermite (see [3]): u6 v6 +5u2v2(u2 v2)+4uv(1 u4v4) = 0 (4) − − − 1/4 1/4 where u = kr and v = k25r . The function kr is also connected to theta functions from the relations 2 θ (q) ∞ 2 ∞ 2 k = 2 , where θ (q)= θ = q(n+1/2) and θ (q)= θ = qn r θ2(q) 2 2 3 3 3 n= n= X−∞ X−∞ (5) 1 π√r q = e− . -
Symmetric Functions Over Finite Fields
Symmetric functions over finite fields Mihai Prunescu ∗ Abstract The number of linear independent algebraic relations among elementary symmetric poly- nomial functions over finite fields is computed. An algorithm able to find all such relations is described. The algorithm consists essentially of Gauss' upper triangular form algorithm. It is proved that the basis of the ideal of algebraic relations found by the algorithm consists of polynomials having coefficients in the prime field Fp. A.M.S.-Classification: 14-04, 15A03. 1 Introduction The problem of interpolation for symmetric functions over fields is an important question in Algebra, and a lot of aspects of this problem have been treated by many authors; see for example the monograph [2] for the big panorama of symmetric polynomials, or [3] for more special results concerning the interpolation. In the case of finite fields the problem of interpolation for symmetric functions is in the same time easier than but different from the general problem. It is easier because it can always be reduced to systems of linear equations. Indeed, there are only finitely many monomials leading to different polynomial functions, and only finitely many tuples to completely define a function. The reason making the problem different from the general one is not really deeper. Let Si be the elementary symmetric polynomials in variables Xi (i = 1; : : : ; n). If the exponents are bounded by q, one has exactly so many polynomials in Si as polynomials in Xi, but strictly less symmetric functions n from Fq ! Fq than general functions. It follows that a lot of polynomials in Si must express the constant 0 function. -
Notes for a Talk on Resolvent Degree
Roots of topology Tobias Shin Abstract These are notes for a talk in the Graduate Student Seminar at Stony Brook. We discuss polynomials, covering spaces, and Galois theory, and how they all relate through the unifying concept of \resolvent degree", following Farb and Wolfson[1]. We will also see how this concept relates Hilbert's 13th problem (among others) to classical enumerative problems in algebraic geometry, such as 27 lines on a smooth cubic, 28 bitangents on a planar quartic, etc. Algebraic functions and roots of topology First, a historial note: the motivation for the concept of the fundamental group comes from studying differential equations on the complex plane (i.e., the theory of Riemann surfaces); namely, in the study of the monodromy of multi-valued complex functions. Consider for example, the path integral R 1=z around the punctured plane. This is nonzero and in fact, its value changes by multiples of 2πi based on how many times you wind around the puncture. This behavior arises since the path integral of this particular function is multivalued; it is only well defined as a function up to branch p cuts. In a similar spirit, we can consider the function z which, as we go around the punctured plane once, sends a specified value to its negative. More specifically, suppose we have a branched covering space π : Y ! X, i.e., a map and a pair of spaces such that away from a nowhere dense subset of X, called the branched locus of X, we have that π is a covering map. -
Introduction to Symmetric Functions Chapter 3 Mike Zabrocki
Introduction to Symmetric Functions Chapter 3 Mike Zabrocki Abstract. A development of the symmetric functions using the plethystic notation. CHAPTER 2 Symmetric polynomials Our presentation of the ring of symmetric functions has so far been non-standard and re- visionist in the sense that the motivation for defining the ring Λ was historically to study the ring of polynomials which are invariant under the permutation of the variables. In this chapter we consider the relationship between Λ and this ring. In this section we wish to consider polynomials f(x1, x2, . , xn) ∈ Q[x1, x2, . , xn] such that f(xσ1 , xσ2 , ··· , xσn ) = f(x1, x2, . , xn) for all σ ∈ Symn. These polynomials form a ring since clearly they are closed under multiplication and contain the element 1 as a unit. We will denote this ring Xn Λ = {f ∈ Q[x1, x2, . , xn]: f(x1, x2, . , xn) =(2.1) f(xσ1 , xσ2 , . , xσn ) for all σ ∈ Symn} Xn Pn k Now there is a relationship between Λ and Λ by setting pk[Xn] := k=1 xi and define a map Λ −→ ΛXn by the linear homomorphism (2.2) pλ 7→ pλ1 [Xn]pλ1 [Xn] ··· pλ`(λ) [Xn] with the natural extension to linear combinations of the pλ. In a more general setting we will take the elements Λ to be a set of functors on polynomials k pk[xi] = xi and pk[cE + dF ] = cpk[E] + dpk[F ] for E, F ∈ Q[x1, x2, . , xn] and coefficients c, d ∈ Q then pλ[E] := pλ1 [E]pλ2 [E] ··· pλ`(λ) [E]. This means that f ∈ Λ will also be a Xn function from Q[x1, x2, . -
Worksheet on Symmetric Polynomials
Worksheet on Symmetric Polynomials Renzo's Math 281 December 3, 2008 A polynomial in several variables x1; : : : ; xn is called symmetric if it does not change when you permute the variables. For example: 1. any polynomial in just one variable is symmetric because there is nothing to permute... 2 2 2. the polynomial x1 + x2 + 5x1 + 5x2 − 8 is a symmetric polynomial in two variables. 3. the polynomial x1x2x3 is a symmetric polynomial in three variables. 2 4. the polynomial x1 + x1x2 is NOT a symmetric polynomial. Permuting x1 2 and x2 you obtain x2 + x1x2 which is different! Problem 1. Familiarize yourself with symmetric polynomials. Write down some examples of symmetric polynomials and of polynomials which are not sym- metric. Problem 2. Describe all possible symmetric polynomials: • in two variables of degree one; • in three variables of degree one; • in four variables of degree one; • in two variables of degree two; • in three variables of degree two; • in four variables of degree two; • in two variables of degree three; • in three variables of degree three; • in four variables of degree three. 1 Now observe the set of symmetric polynomials in n variables. It is a vector subspace of the vector space of all possible polynomials. (Why is it true?) In particular we want to think about Q-vector subspaces...this simply means that we only allow coefficients to be rational numbers, and that we allow ourselves to only stretch by rational numbers. This is VERY IMPORTANT for our purposes, but in practice it doesn't change anything in terms of how you should go about these probems. -
Homeomorphism of S^ 1 and Factorization
HOMEOMORPHISMS OF S1 AND FACTORIZATION MARK DALTHORP AND DOUG PICKRELL Abstract. For each n> 0 there is a one complex parameter family of home- omorphisms of the circle consisting of linear fractional transformations ‘con- jugated by z → zn’. We show that these families are free of relations, which determines the structure of ‘the group of homeomorphisms of finite type’. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup fac- torization is related to triangular factorization (i.e. conformal welding), and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients. 0. Introduction In this paper we consider the question of whether it is possible to factor an orientation preserving homeomorphism of the circle, belonging to a given group, as a composition of ‘linear fractional transformations conjugated by z zn’. What we mean by factorization depends on the group of homeomorphisms we→are considering. In the introduction we will start with the simplest classes of homeomorphisms and build up. For algebraic homeomorphisms, factorization is to be understood in terms of generators and relations. For less regular homeomorphisms factorization involves limits and ordering, and in particular is highly asymmetric with respect to inversion. 0.1. Diffeomorphisms of Finite Type. Given a positive integer n and wn ∆ := w C : w < 1 , define a function φ : S1 S1 by ∈ { ∈ | | } n → n 1/n (1+w ¯nz− ) (0.1) φn(wn; z) := z n 1/n , z =1 (1 + wnz ) | | arXiv:1408.5402v4 [math.GT] 22 Jun 2019 1 It is straightforward to check that φn Diff(S ), the group of orientation pre- 1 1∈ serving diffeomorphisms of S , and φ− (z) = φ ( w ; z). -
On Generalized Integrals and Ramanujan-Jacobi Special Functions
On Generalized Integrals and Ramanujan-Jacobi Special Functions N.D.Bagis Department of Informatics, Aristotele University Thessaloniki, Greece [email protected] Keywords: Elliptic Functions; Ramanujan; Special Functions; Continued Fractions; Generalization; Evaluations Abstract In this article we consider new generalized functions for evaluating integrals and roots of functions. The construction of these general- ized functions is based on Rogers-Ramanujan continued fraction, the Ramanujan-Dedekind eta, the elliptic singular modulus and other similar functions. We also provide modular equations of these new generalized functions and remark some interesting properties. 1 Introduction Let ∞ η(τ)= eπiτ/12 (1 e2πinτ ) (1) − n=1 Y denotes the Dedekind eta function which is defined in the upper half complex plane. It not defined for real τ. Let for q < 1 the Ramanujan eta function be | | ∞ arXiv:1309.7247v3 [math.GM] 15 Nov 2015 f( q)= (1 qn). (2) − − n=1 Y The following evaluation holds (see [12]): 1/3 1/2 1/24 1/12 1/3 1/2 f( q)=2 π− q− k k∗ K(k) (3) − 2 where k = kr is the elliptic singular modulus, k∗ = √1 k and K(x) is the complete elliptic integral of the first kind. − The Rogers-Ramanujan continued fraction is (see [8],[9],[14]): q1/5 q1 q2 q3 R(q) := ... (4) 1+ 1+ 1+ 1+ 1 which have first derivative (see [5]): 1 5/6 4 6 5 5 R′(q)=5− q− f( q) R(q) R(q) 11 R(q) (5) − − − − and also we can write p dR(q) 1 1/3 2/3 6 5 5 =5− 2 (kk∗)− R(q) R(q) 11 R(q) . -
Differential Algebra and Liouville's Theorem
Math 4010/5530 Elementary Functions and Liouville's Theorem April 2016 Definition: Let R be a commutative ring with 1. We call a map @ : R ! R a derivation if @(a+b) = @(a)+@(b) and @(ab) = @(a)b + a@(b) for all a; b 2 R. A ring equipped with a particular derivative is called a differential ring. For convenience, we write @(a) = a0 (just like in calculus). If R is a differential ring and an integral domain, it is a differential integral domain. If R is a differential ring and a field, it is a differential field. Let R be a differential ring. • 10 = (1·1)0 = 10 ·1+1·10 so 10 = 10 +10 and so 0 = 10. Notice that @ is a group homomorphism (consider R as an abelian group under addition), so @(na) = n@(a) for all a 2 R and n 2 Z. Therefore, @(n1) = n@(1) = n0 = 0. Thus everything in the prime subring is a constant (i.e. has derivative zero). • Given a 2 R, notice that (a2)0 = (a · a)0 = a0a + aa0 = 2aa0. In fact, we can prove (using induction), that n n−1 0 a = na a for any n 2 Z≥0 (The power rule) • Let a 2 R× (a is a unit). Then 0 = 10 = (aa−1)0 = a0a−1 + a(a−1)0 so a(a−1)0 = −a−1a0. Dividing by a, we get that (a−1)0 = −a−2a0. Again, using induction, we have that an = nan−1 for all n 2 Z (if a−1 exists). -
Abel‒Ruffini Theorem
AbelRuffini theorem 1 Abel–Ruffini theorem In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.[1] Interpretation The content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equations are unsolvable. In fact, the opposite is true: every non-constant polynomial equation in one unknown, with real or complex coefficients, has at least one complex number as solution; this is the fundamental theorem of algebra. Although the solutions cannot always be expressed exactly with radicals, they can be computed to any desired degree of accuracy using numerical methods such as the Newton–Raphson method or Laguerre method, and in this way they are no different from solutions to polynomial equations of the second, third, or fourth degrees. The theorem only concerns the form that such a solution must take. The theorem says that not all solutions of higher-degree equations can be obtained by starting with the equation's coefficients and rational constants, and repeatedly forming sums, differences, products, quotients, and radicals (n-th roots, for some integer n) of previously obtained numbers. This clearly excludes the possibility of having any formula that expresses the solutions of an arbitrary equation of degree 5 or higher in terms of its coefficients, using only those operations, or even of having different formulas for different roots or for different classes of polynomials, in such a way as to cover all cases. (In principle one could imagine formulas using irrational numbers as constants, but even if a finite number of those were admitted at the start, not all roots of higher-degree equations could be obtained.) However some polynomial equations, of arbitrarily high degree, are solvable with such operations.