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RF21 Freeware Polynomials   (Site not responding. Last check: 2007-11-02) The converse problem of finding a polynomial having a given set of roots, is also supported, for up to 4 complex roots or an unlimited number of real roots. For polynomials of grad 5 and up it was shown by Abel in 1826 and later Galois that there can be no similar formulae solving the general case. There is a whole theory on symmetric functions, centered around symmetric polynomials. members.aol.com /rf24exe/polynom.html   (1276 words)

Springer Online Reference Works If the polynomial is not identically zero, then among the terms with non-zero coefficients (it is assumed that similar terms have been reduced) there is at least one of highest degree: this highest degree is called the degree of the polynomial. A polynomial of which all terms have the same degree is called a homogeneous polynomial or a form; forms of the first, second or third degree are called linear, quadratic or cubic, and, according to the number of variables (two or three), they are called dyadic (binary) or triadic (ternary) (for example, A polynomial which can be represented as a product of polynomials of smaller degree with coefficients from a given field is called reducible (over that field); otherwise it is called irreducible. eom.springer.de /p/p073690.htm   (987 words)

Polynomial Summary For example, Z[x] is the subring of polynomials with coefficients in the ring Z of integers, Q[x] is the subring of polynomials with coefficients in the field Q of rational numbers, and ℜ[x] is the subring of polynomials with coefficients in the field ℜ of real numbers. Usually a polynomial function is simply called a polynomial, and it is clear from the context if the polynomial is to be regarded as a function or as an element of a ring of polynomials. Polynomials are classified by their degree and number of variables. www.bookrags.com /Polynomial   (3406 words)

Newton's identities - Wikipedia, the free encyclopedia Frequently, this polynomial is regarded as the characteristic polynomial of a linear operator or matrix; then the roots are called eigenvalues. We can obtain "finer" decompositions by writing general symmetric polynomials as sums of homogeneous polynomials; that is, a symmetric polynomial in which all the terms have the same degree. The resulting Gröbner basis of an ideal in a ring of multivariable polynomials is analogous to a vector basis for a subspace of vector space, and is ideally suited for computations involving ideals in polynomial rings, which is the basis concept of algebraic geometry. en.wikipedia.org /wiki/Newton's_identities   (1734 words)

Symmetric function - Wikipedia, the free encyclopedia In mathematics, a symmetric function of multiple variables is one that is invariant under permutation of its variables. The theory of symmetric polynomials is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. The polynomial relations underlying that assertion are universal (independent of choice of P); and, if we work with the symmetric polynomials created from a monomial, we can eliminate dependence on K, too, to get formulae with integer coefficients. en.wikipedia.org /wiki/Symmetric_function   (543 words)

Symmetric Pseudoprimes We could also define symmetric pseudoprimes in terms of Newton's sums for the roots of a polynomial.  Let f denote a monic polynomial of degree d with integer coefficients, and let s(k) denote the sum of the kth powers of the roots of f.  Lucas observed that if p is a prime then s(p Symmetric pseudoprimes tend to be more rare relative to polynomials with larger Galois groups. In the articles listed below, basic propositions and computational techniques associated with symmetric pseudoprimes are presented, along with specific examples relative to selected polynomials of degrees 1 to 5, and the final article describes complete congruence conditions on the terms of arbitrary linear recurring sequences. www.mathpages.com /home/kmath003/kmath003.htm   (379 words)

PlanetMath: symmetric polynomial Every symmetric polynomial can be written as a polynomial expression in the elementary symmetric polynomials. Cross-references: elementary symmetric polynomials, permutation, symmetric, ring, variables, polynomial This is version 3 of symmetric polynomial, born on 2002-01-05, modified 2004-02-02. planetmath.org /encyclopedia/SymmetricPolynomial.html   (71 words)

PlanetMath: elementary symmetric polynomial The elementary symmetric polynomials can also be constructed by taking the sum of all possible degree elementary symmetric polynomial in terms of power sums This is version 5 of elementary symmetric polynomial, born on 2002-01-05, modified 2006-10-22. planetmath.org /encyclopedia/ElementarySymmetricPolynomial.html   (45 words)

SymmetricPolynomials The package provides functions for generating elementary symmetric polynomials and for representing symmetric polynomials in terms of elementary symmetric polynomials. Here is the elementary symmetric polynomial of degree three in four variables. Here the elementary symmetric polynomials in the symmetric part of the input polynomial are replaced with the given variables. documents.wolfram.com /v5/Add-onsLinks/StandardPackages/Algebra/SymmetricPolynomials.html   (180 words)

Introduction Since the partitions of weight n prescribe the symmetric function basis elements of degree n which generate Lambda^n, (the symmetric polynomials in n indeterminates), then the number of indeterminates being used will normally be equal to the degree of the symmetric functions considered. The monomial symmetric function m_lambda is the sum over the orbits of x^lambda under the action of the symmetric group. While the degree of the symmetric function fixes the degree of the resulting polynomial, the number of indeterminates is arbitrary. wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text1330.htm   (940 words)

Springer Online Reference Works with a unit, which is a symmetric function in its variables, that is, is invariant under all permutations of the variables: The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. A skew-symmetric, or alternating, polynomial is a polynomial eom.springer.de /s/s091700.htm   (333 words)

Invariants under Actions An example of symmetric polynomials arises in looking at general polynomials and their roots. 314 of the IVA is stated in rather complicated and obscure terms, but its main consequence is that it provides a good way to test whether an arbitrary polynomial is symmetric, and gives a somewhat quicker way to decompose such a polynomial into polynomials in the elementary functions. To say that a polynomial  g  is homogeneous of total degree k means that every term appearing in  g  has total degree k. www.math.ohio-state.edu /~ault/Papers/invariants.html   (1043 words)

Creation Symmetric functions are symmetric polynomials over some coefficient ring, hence the algebra of symmetric functions is defined by specifying this ring. General symmetric functions are linear combinations of basis elements and can be created by taking such linear combinations or via coercion from either another basis or directly from a polynomial. Given a multivariate polynomial f which is symmetric in all of its indeterminates (and hence is a symmetric function), return f as an element of the algebra of symmetric functions A. This element returned will be expressed in terms of the symmetric function basis of A. Example www.math.lsu.edu /magma/text1365.htm   (900 words)

Symmetric Polynomials   (Site not responding. Last check: 2007-11-02) Given a polynomial ring P of rank n, and an integer k with 1 <= k <= n, return the k-th elementary symmetric polynomial of P. IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt Given a polynomial f from a polynomial ring P of rank n, return whether f is a symmetric polynomial of P (i.e., is symmetric in all the n variables of P). We create a symmetric polynomial from Q[a, b, c, d] and express it in terms of the elementary symmetric polynomials. www.umich.edu /~gpcc/scs/magma/text599.htm   (99 words)

Springer Online Reference Works is a quasi-symmetric polynomial in three variables that is not symmetric. of quasi-symmetric polynomials in finite of countably many indeterminates, which are the quasi-symmetric power series of bounded degree. There is a completely different notion in the theory of functions of a complex variable that also goes by the name quasi-symmetric function; cf., e.g., [a7]. eom.springer.de /Q/q120060.htm   (393 words)

2.0 Symmetric Pseudoprimes Relative to Selected Polynomials By Proposition 6, a composite integer N co-prime to 6 is a symmetric pseudoprime relative to the third degree polynomial Perrin's polynomial is the fundamental reversible cubic with discriminant -23 (which is the smallest negative discriminant for a cubic).  As explained by Shanks, et al, there are 12 reversible cubics with discriminant -23, but the corresponding recurrences are all of the form s(k) = s A composite integer N co-prime to 120 is a symmetric pseudoprime relative to the fifth degree polynomial www.mathpages.com /home/kmath003/part2/sympart2.htm   (2751 words)

Symmetric Polynomial Solutions of the Generalised Laplace's Equation We examine the problem of finding symmetric homogeneous polynomial solutions of the generalized Laplace's equation which arises in the context of Calogero-Sutherland Moser models. We suggest that Jack polynomials might provide the best basis for this purpose as the action of the generalised Laplacian on Jack Polynomials can explicitly be written down in terms of generalised binomial coefficients defined with respect to Jack Polynomials. Explicit expressions for the generalised binomial coefficients for partitions of N for N less than equal to six are presented. www.maths.uq.edu.au /cmp/Seminars/announcements/2000/sc18072000.html   (109 words)

ECCC Report TR01-035 and related Papers   (Site not responding. Last check: 2007-11-02) This new model is related to standard models=20 of arithmetic circuits, especially to depth 3 circuits. The first lower bound is for computing the determinant, and the second is for computing the sum of two monomials.=20 The main technical contribution relates the maximal dimension=20 of linear subspaces on which $S_{m}^d$ vanishes, and lower=20 bounds to the symmetric model. Using our techniques we also prove quadratic lower bounds for depth 3 circuits computing the elementary symmetric polynomials=20 of degree $alpha n$ (where $0< alpha < 1$ is a constant), thus extending the result of cite{SW}.=20 These are the best lower bounds known for depth 3 circuits over fields of characteristic zero. eccc.hpi-web.de /eccc-reports/2001/TR01-035/index.html   (211 words)

Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-11-02) Date: 07/31/97 at 11:36:48 From: Doctor Wilkinson Subject: Re: Symmetric polynomials A symmetric polynomial is one that doesn't change when you permute its variables. For example, x^2 + y^2 is a symmetric polynomial in x and y, because if you exchange x and y, the polynomial doesn't change; but x^2 + 2y^2 isn't symmetric, because if you exchange x and y you get 2x^2 + y^2, which is different. The elementary symmetric polynomials in x1, x2, x3, x4,..., xn are s1 = x1 + x2 + x3 +... mathforum.org /library/drmath/view/56411.html   (233 words)

Symmetric Matrix Polynomial Equation: Interpolation Results - Henrion, Sebek (ResearchIndex)   (Site not responding. Last check: 2007-11-02) Abstract: A new numerical procedure is proposed to solve the symmetric matrix polynomial equation A T (\Gammas)X (s) + X T (\Gammas)A(s) = 2B(s) that is frequently encountered in control and signal processing. It is based on interpolation and takes fully advantage of symmetry of the equation by reducing the original problem dimension. 12 Polynomial and Rational Matrix Interpolation : Theory and Co.. citeseer.ist.psu.edu /henrion97symmetric.html   (513 words)

Elliptic Curves A polynomial is symmetric when any of the variables can be exchanged without altering the polynomial. If we have any symmetric polynomial, it can be expressed as a combination of the elementary symmetric polynomials. where g(x) is a polynomial of degree 3, with roots, a, b and c (not to get confused with the coefficients with uppercase lettering). www.willamette.edu /~zizza/Courses/SeniorSeminar/G2.2/ECurves.html   (1686 words)

Polynomial algebra Gröbner bases can be used to solve systems of polynomial equations, and are also useful for other applications such as integer programming and converting parametric curves and surfaces to implicit form. Note: The “content” of a polynomial is the gcd of its coefficients. cyclotomic polynomial is the minimal polynomial of the primitive n www.ibiblio.org /technicalc/packages/mathtools/polynomials.htm   (609 words)

Math/CS Chats   (Site not responding. Last check: 2007-11-02) A polynomial in several variables is said to be "symmetric" if any permutation of its variables leaves the polynomial unchanged (e.g. In other words, a polynomial in n variables is symmetric if it remains fixed under the permutation action of the symmetric group S It is a classical fact that any symmetric polynomial can be written as a polynomial in "elementary symmetric polynomials", and the proof of this is given by a constructive algorithm. www.dickinson.edu /~braught/chats/fall02chats.html   (601 words)

Symmetric and non-symmetric quantum Capelli polynomials   (Site not responding. Last check: 2007-11-02) In "Difference operators and symmetric functions defined by their zeros" (joint with Siddhartha Sahi) we introduced a new family of symmetric polynomials which are related to Capelli identities. The purpose of this paper is twofold: I quantize the construction and obtain symmetric polynomials depending on two parameters q and t. Integrality of two variable Kostka functions are generalized by obtaining integrality results for the quantum Capelli polynomials. www.math.rutgers.edu /~knop/papers/QC.html   (104 words)

Abstracts of Malika More Let A be some L-sentence and P be a polynomial of degree k of Z[X] asymtotically greater than or equal to identity function on N. We produce a sentence P(A) representing P(S(A)), i.e. When the language is given by a deterministic finite automaton, deciding whether a regular language is symmetric is polynomial. We prove that, if the language is given by a deterministic finite automaton, deciding whether a symmetric regular language is affine or bijunctive, that is whether the corresponding satisfiability problem is in P, is polynomial. www.math.unicaen.fr /~more/recherche/abstract2.html   (1271 words)

ECCC Report TR02-052 and related Papers   (Site not responding. Last check: 2007-11-02) Abstract: Elementary symmetric polynomials $S_n^k$ are used as a benchmark for the bounded-depth arithmetic circuit model of computation. In this work we prove that $S_n^k$ modulo composite numbers $m=p_1p_2$ can be computed with much fewer multiplications than over any field, if the coefficients of monomials $x_{i_1}x_{i_2}cdots x_{i_k}$ are allowed to be 1 either mod $p_1$ or mod $p_2$ but not necessarily both. Moreover, the number of multiplications remain sublinear while $k=O(loglog n).$ In contrast, the well-known Graham-Pollack bound yields an $n-1$ lower bound for the number of multiplications even for the second elementary symmetric polynomial $S_n^2$. eccc.hpi-web.de /eccc-reports/2002/TR02-052/index.html   (231 words)

Module for Graeffe's Method Approximate the roots of the following polynomials using the separated root theorem. The heart of the Graeffe method is to start with "mildly" separated roots and construct a related polynomial with sufficiently widely separated roots. So it is an unnecessary step to form the polynomials. math.fullerton.edu /mathews/n2003/GraeffeMethodMod.html   (524 words)

Springer Online Reference Works consisting of all even symmetric polynomials in the Wu generators. The characteristic class determined by an even symmetric polynomial in the Wu generators can be expressed in Pontryagin classes as follows. First, the polynomial is written in elementary symmetric functions of the variables eom.springer.de /p/p073750.htm   (732 words)

Thesis   (Site not responding. Last check: 2007-11-02) Homotopy continuation methods have proven to be reliable for computing numerically approximations to all isolated solutions of polynomial systems. This root counting method proved to be more flexible and effective, especially for symmetric polynomial systems. Hereby we have provided practical evidence that the investigated homotopy continuation methods are effective in solving polynomial systems. www.math.uic.edu /~jan/thesis.html   (371 words)

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