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	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)

	Symmetric polynomial - Wikipedia, the free encyclopedia
		In mathematics, a symmetric polynomial is a polynomial in n variables P(X
		They are the building blocks for all symmetric polynomials in these variables, meaning that any symmetric polynomial in n variables can be obtained from the elementary symmetric polynomials via several multiplications and additions.
		Symmetric polynomials are important to linear algebra, representation theory, and Galois theory.
	en.wikipedia.org /wiki/Symmetric_polynomial (196 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 (547 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)

	Polynomial Summary
		The degree of a term in a polynomial is the sum of all of the exponents on the variables in that term, where a variable with no exponent is understood to have an exponent of 1.
		In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one unknown.
		In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants.
	www.bookrags.com /Polynomial (3406 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		In mathematics, a symmetric function of multiple variables is one that is invariant under permutation of its variables; that is, the value of the function does not depend on the order of the n-tuple of arguments.
		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.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=symmetric_function (493 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.
	mathpages.com /home/kmath003/kmath003.htm (379 words)

	RF21 Freeware Polynomials
		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.
		elementary symmetric polynomial in the n unknowns a
	members.aol.com /rf24exe/polynom.html (1276 words)

	WWW interactive multipurpose server
		OEF polynomial, collection of exercises on polynomials of one variable (real or complex coefficients).
		Polynomial sweep, graphs and roots of a polynomial, with animated deformation.
		Symmetric split, write a given matrix as sum of symmetric and antisymmetric matrices.
	www.eval-wims.com /wims/wims.cgi (2370 words)

	Wolfram Research, Inc.
		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 /v3/AddOns/Alg_SymmetricPolynomials-.html (176 words)

	3.
		A symmetric polynomial can be represented in one and only one manner as the sum of homogeneous symmetric polynomials of different degrees.
		A homogeneous polynomial is transformed by a permutation of the indeterminates into a homogeneous polynomial of the same degree, whence each of the homogeneous portions of a symmetric polynomial are transformed each themselves by every permutation and they are therefore symmetric homogeneous polynomials.
		When the characteristic of K is equal to 2, every alternating polynomial is symmetric and conversely; when the characteristic differs from 2, the polynomial 0 is the only polynomial which can be considered to be symmetric as well as alternating.
	kr.cs.ait.ac.th /~radok/math/mat5/algebra32.htm (5560 words)

	Algebra - Numericana
		Symmetric polynomials of 3 variables: Obtain the value of one from 3 others.
		Any symmetrical polynomial of such roots is also a polynomial in U,V,W, and its value may thus be obtained without solving the cubic equation.
		Such expressions of power-sums in terms of the elementary symmetric polynomials are known as Girard-Waring expansions (published in 1629 by Albert Girard and between 1762 and 1782 by Edward Waring).
	home.att.net /~numericana/answer/algebra.htm (4230 words)

	Symmetric Polynomials (Site not responding. Last check: )
		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).
	www.math.ufl.edu /help/magma/text362.html (99 words)

	Re: Computational Invariant Theory, Anyone?
		Symmetric Polynomials (continued) I wrote: > A very important feature of R which we have slurred over for far too > long is that this is a -graded ring-.
		The polynomials in each basis are of course -linearly independent- but in general they will not be -algebraically independent-; this was illustrated by my discussion of the sequence of syzygy ideals.
		For each partition, we define a polynomial in the following ways: (i) the -monomial symmetric polynomial- m[d1,d2,..dn](x1,x2,..xn) is the sum of all monomials of "shape" x1^d1 x2^d2..
	www.lns.cornell.edu /spr/2002-05/msg0041674.html (625 words)

	[No title] (Site not responding. Last check: )
		Furthermore, since all of the factors in the term that contribute to the x^(p-1) term of polynomial (2) are x, none of the factors are constants, so the coefficient of said term, a_0, is 1.
		Thus, the constant term of the polynomial (2), a_(p-1), is (-1)(-2)...(-(p-1)) = (-1)^(p-1) (p-1)!.
		Note that this is precisely the definition of e_k(-1, -2,..., -(p-1)) where e_k is the k-th elementary symmetric polynomial.
	www.cs.berkeley.edu /~daw/teaching/cs70-f03/chocolate.txt (750 words)

	Elementary symmetric polynomial - Definition, explanation
		In mathematics, elementary symmetric polynomials are basic building block for symmetric polynomials.
		Note that these polynomials are indeed symmetric, as when some variables are interchanged, the polynomials stay the same.
		The uses of these polynomials are described in the symmetric polynomials article.
	www.calsky.com /lexikon/en/txt/e/el/elementary_symmetric_polynomial.php (132 words)

	Symmetric Polynomials
		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.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1175.htm (99 words)

	Basic facts
		The first method involves symmetric polynomials, which are interesting enough in their own right that I'll discuss them in some detail.
		be a symmetric polynomial (with integer coefficients) in
		By the theorem, this polynomial is a polynomial in the elementary symmetric functions of the
	mathcircle.berkeley.edu /BMC3/bamc/node1.html (496 words)

	Selected Matches for: Author/Related=valibouze
		In $§4$ the study of the characteristic polynomial in the algebra of universal decomposition reminds one of the methods of Arnaudiès (1992), J.-L. Lagrange (1770) and A. Cauchy (1882).
		The resolvent of $f(x)$ by $h(x_1,\cdots,x_n)$ is the univariate polynomial $(y-h_1(a_1,\cdots,a_n))\cdots (y-h_e(a_1, \cdots,a_n))$.
		In this paper it is shown (after a technical definition of the concept) that "transforming polynomial equations by a morphism" is algorithmically equivalent to elementary elimination theory by resultants and also to making change of bases for symmetric polynomials.
	www-calfor.lip6.fr /~avb/Bibliographies/konquerorEjMOrb.html (2903 words)

	[No title] (Site not responding. Last check: )
		The first lower bound is for computing the determinant, and the second is for computing the sum of two monomials.
		The main technical contribution relates the maximal dimension of linear subspaces on which $S_{m}^d$ vanishes, and lower bounds to the symmetric model.
		Using our techniques we also prove quadratic lower bounds for depth 3 circuits computing the elementary symmetric polynomials of degree $\alpha n$ (where $0< \alpha < 1$ is a constant), thus extending the result of \cite{SW}.
	www.cs.huji.ac.il /~theorys/2001/Shpilka_Amir (164 words)

	Proc.Inst.Statist.Math.49-1
		Recently, an algorithm for expressing zonal polynomials of arbitrary order in terms of elementary symmetric polynomials has been proposed by Hashiguchi, the second author of this article, and his coworkers including the third author.
		(Y) denote the zonal polynomial and the elementary symmetric polynomial, respectively, identified with a partition \kappa of which length is not greater than p and a p × p symmetric matrix Y of independent variables.
		A family of probability density functions with the part consisting of the sum of a main variable and its reciprocal is introduced as a family of symmetric reciprocal distributions.
	www.ism.ac.jp /editsec/toukei/abstract/49-1e.html (1783 words)

	ECCC Report TR02-052 and related Papers
		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)

	CJM - Elementary Symmetric Polynomials in Numbers of Modulus 1
		CJM - Elementary Symmetric Polynomials in Numbers of Modulus 1
		Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus.
		Elementary Symmetric Polynomials in Numbers of Modulus 1
	journals.cms.math.ca /cgi-bin/vault/view/cartwright1009 (132 words)

	Search ScienceWorld
		Any symmetric polynomial (respectively, symmetric rational function) can be expressed as a polynomial (respectively, rational function) in the elementary symmetric polynomials on those variables.There is a generalization of this theorem to polynomial invariants of permutation groups G, which states that a
		In practice, the vertical offsets from a line (polynomial, surface, hyperplane, etc.) are almost always minimized instead of the perpendicular offsets.
		A technical mathematical object defined in terms of a polynomial ring of n variables over a field k.
	scienceworld.wolfram.com /search/index.cgi?num=&q=Polynomial&start=240 (496 words)

	WWW interactive multipurpose server
		Contrib Cochise 2001, exercises on elementary algebra by Math Dept of Cochise College.
		Contrib Cochise 2000, exercises on elementary algebra by Math Dept of Cochise College.
		Primpoly, search for primitive polynomials over a finite field.
	wims.unice.fr /wims/en_home.html (2494 words)

	The Math Forum - Math Library - Polynomials
		Mathematicians once spent time on a subject call the "Theory of equations," which was full of algorithms and the theory of polynomials and their roots.
		Enter generating polynomials for the ideal I to calculate its Gröbner basis, minimal Gröbner basis, or reduced Gröbner basis, which can be used for finding the solution to simultaneous polynomial equations and for determining the existence...more>>
		Given a monic polynomial f with integer coefficients, a symmetric pseudoprime relative to f is defined as a composite integer N such that every elementary symmetric function of...more>>
	mathforum.org /library/topics/polynomials (1944 words)

	The Limit Behavior of Elementary Symmetric Polynomials of i.i.d. Random Variables When Their Order Tends to Infinity, ...
		The Limit Behavior of Elementary Symmetric Polynomials of i.i.d.
		Let $\xi_1,\xi_2\ldots$ be a sequence of i.i.d.random variables, and consider the elementary symmetric polynomial $S ^(k)(n)$ of order $k =k(n)$ of the first $n$ elements $\xi_1\ldots,\xi_n$ of this sequence.
		The proof is based on the saddlepoint method and a limit theorem for sums of independent random vectors which mayhave some special interest in itself.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aop/1022874824 (228 words)

	[No title] (Site not responding. Last check: )
		The expression you write down for the total is called the kth elementary symmetric polynomial in the n variables s_1,..., s_n.
		The highest of the symmetric polynomials is the product, s_1 s_2...
		+ s_n t_2 = s_1^2 + s_2^2 +...+ s_n^2 t_3 = s_1^3 + s_2^3 +...+ s_n^3 and so on, then the elementary symmetric polynomials may all be expressed in terms of these quantities.
	www.math.niu.edu /~rusin/known-math/97/countme (416 words)

	Math Forum Discussions
		> > neat as the case for elementary symmetric polynomials.
		Among polynomials which are homogeneous of degree 2 in x and 1 in y,
		From the elementary symmetric polynomials and the new one discovered in
	www.mathforum.org /kb/thread.jspa?messageID=490033&tstart=0 (594 words)

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