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Topic: Polynomial ring

	Polynomial - Wikipedia (Site not responding. Last check: 2007-10-01)
		Polynomials are important because they are the simplest functions: their definition involves only addition and multiplication (since the powers are just shorthands for repeated multiplications).
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Weierstrass approximation theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		Formation of the polynomial ring, together with forming factor rings by factoring out ideals[?], are important tools for constructing new rings out of known ones.
	wikipedia.findthelinks.com /po/Polynomial.html (1441 words)

	Polynomial - Wikipedia, the free encyclopedia
		The derivative of a polynomial is a polynomial
		The integral of a polynomial is a polynomial
		In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants.
	en.wikipedia.org /wiki/Polynomial (2691 words)

	Graded Polynomial Rings
		Given a polynomial f of the graded polynomial ring P, this function returns the weighted degree of f, which is the maximum of the weighted degrees of all monomials that occur in f.
		Given a polynomial f of the graded polynomial ring P, this function returns whether f is homogeneous with respect to the weights on the variables of P (i.e., whether the weighted degrees of the monomials of f are all equal).
		Given an ideal I of the graded polynomial ring P, this function returns whether I is homogeneous with respect to the weights on the variables of P (i.e., whether I possesses a basis consisting of homogeneous polynomials alone).
	www.umich.edu /~gpcc/scs/magma/text848.htm (952 words)

	13: Commutative rings and algebras
		Of particular interest are several classes of rings of interest in number theory, field theory, algebraic geometry, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.
		Conversely, the study of a ring is often focused by the examination of related fields, such as the quotients by each of the maximal ideals, or, in the case of integral domains, by the quotient field.
		Rings associated to group a group G shed light on the structure of G, particular rings of invariants k(V)^G (given a group action on a vector space V), cohomology rings H^(G,Z), group rings* Z[G], and representation rings R(G).
	www.math.niu.edu /~rusin/known-math/index/13-XX.html (2760 words)

	PlanetMath: polynomial ring over integral domain
		The theorem may by induction be generalized for the polynomial ring
		"polynomial ring over integral domain" is owned by pahio.
		This is version 4 of polynomial ring over integral domain, born on 2005-03-30, modified 2006-09-29.
	planetmath.org /encyclopedia/CoefficientRing.html (80 words)

	Element Operations (Site not responding. Last check: 2007-10-01)
		Given a multivariate polynomial p with coefficients in R, this function returns a sequence of `base' coefficients, that is, a sequence of elements of R occurring as non-zero coefficients of the monomials in p.
		Given a multivariate polynomial p in P, this function returns a sequence of monomials, that is, a sequence of monomial elements of P occurring with non-zero coefficients in p.
		Given a multivariate polynomial p in P = R[x_1,..., x_n], this function returns a sequence of terms with respect to a given variable v=x_i, that is, the function returns a sequence of elements of P that form the terms (ascending order) of p regarded as a polynomial sum_j c_j x_i^j.
	www.math.ufl.edu /help/magma/text345.html (2034 words)

	More on Polynomial
		Because of their simple structure, polynomials are very easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions.
		Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x.
		To every polynomial f in R[X], one can associate a polynomial function with domain and range equal to R. One obtains the value of this function for a given argument r by everywhere replacing the symbol X in f's expression by r.
	www.artilifes.com /polynomial.htm (1982 words)

	PlanetMath: homogeneous polynomial
		A homogeneous polynomial of degree 1 is called a linear form; a homogeneous polynomial of degree 2 is called a quadratic form; and a homogeneous polynomial of degree 3 is called a cubic form.
		In fact, a homogeneous function that is also a polynomial is a homogeneous polynomial.
		This is version 14 of homogeneous polynomial, born on 2004-12-14, modified 2006-02-24.
	planetmath.org /encyclopedia/PolynomialForm.html (254 words)

	PlanetMath: polynomial ring (Site not responding. Last check: 2007-10-01)
		Similarly, a binomial is a polynomial with exactly two nonzero coefficients, and a trinomial is a polynomial with exactly three nonzero coefficients.
		In any number of variables, a polynomial ring is a graded ring with
		This is version 5 of polynomial ring, born on 2001-10-23, modified 2005-07-24.
	planetmath.org /encyclopedia/PolynomialRing.html (311 words)

	Polynomial ring - Wikipedia, the free encyclopedia
		In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring.
		Every commutative ring that is a finitely-generated algebra over a field can be written as a quotient of a polynomial ring.
		An interesting example of a ring obtained by using polynomials is the ring of Frobenius polynomials, where the ring multiplication is given by function composition, rather than by polynomial multiplication.
	en.wikipedia.org /wiki/Polynomial_ring (837 words)

	Macaulay 2 home page
		On the cohomology ring of the moduli space of Higgs bundles II: Relations in rank 2, by Tamas Hausel and Michael Thaddeus, arXiv:math.AG/0003094.
		The ring of arithmetical functions with unitary convolution: the [n]-truncation, by Jan Snellman, arXiv:math.AC/0208183.
		Conjectures on the ring of commuting matrices, by Freyja Hreinsdottir, arXiv:math.AC/0501465.
	www.math.uiuc.edu /Macaulay2 (4695 words)

	Rings
		A Division Algebra is a nontrivial ring (not necessarily commutative) in which all nonzero elements are invertible.
		The kernel of a homomorphism of rings f: A --> B is the ideal in A consisting of those elements a in A such that f(a) = 0.
		The polynomial ring A[t] over a ring A consists of all the formal polynomials with coefficients from A in an indeterminate symbol t.
	mcraefamily.com /mathhelp/BasicAARings.htm (1006 words)

	[No title]
		In the algebra side, the universal Witt ring W(R) of a commutative ring R is a ~-ring.
		Since a filtered ring R has a topology on it, one is tempted to ask if the moduli set of filtered ~-ring structures on it has a natural topology that is, in some sense, compatible with the topology on R. This is indeed the case, as was shown by the author in [5].
		By a filtered ring, we mean a commutative ring R with unit together with a decreasing sequence of ideals R = I0 I1 I2.
	www.math.purdue.edu /research/atopology/YauD/truncated.txt (3904 words)

	[ref] 64 Polynomials and Rational Functions
		Polynomial rings of smaller rank naturally embed in rings of higher rank; if S is a subring of R then a polynomial ring over S naturally embeds in a polynomial ring over R of the same rank.
		Therefore two univariate polynomials may be considered to be in the same univariate polynomial ring if their indeterminates have the same number or one if of them is constant.
		A polynomial function is an element of a polynomial ring (not necessarily an UFD).
	www.gap-system.org /Manuals/doc/htm/ref/CHAP064.htm (5061 words)

	polynomial rings
		Conversely, an element of the polynomial ring that is known to be a scalar can be lifted back to the coefficient ring.
		When displaying an element of an iterated polynomial ring, parentheses are used to organize the coefficients recursively, which may themselves be polynomials.
		A basis of the subspace of ring elements of a given degree can be obtained in matrix form with basis.
	www.math.temple.edu /computing/Macaulay2/0328.html (687 words)

	Nullstellensatz
		The quotient ring is equal to c, h is maximal, and h = m.
		Thus h is an ideal of polynomials taken from a polynomial ring, and we are interested in the algebraic set h′ in c
		This polynomial is irreducible in the ufd k[x,y,z], hence it is prime.
	www.mathreference.com /ag,nullsatz.html (785 words)

	Polynomial Rings and Polynomials (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-10-01)
		Create a multivariate polynomial ring in n>0 indeterminates over the ring R. The ring is regarded as an R-algebra via the usual identification of elements of R and the constant polynomials.
		Explicit coercion is always allowed between polynomial rings having the same number of variables (and suitable base rings), whether they are global or not, and the coercion maps the i-variable of one ring to the i-th variable of the other ring.
		Given a multivariate polynomial ring P=R[x_1,..., x_n], as well as a polynomial f in a univariate polynomial ring R[x] over the same coefficient ring R, return an element q of P corresponding to f in the indeterminate v=x_i; that is, q in P is defined by q=sum_j f_jx_i^j where f=sum_j f_jx^j.
	www.math.lsu.edu.cob-web.org:8888 /magma/text566.htm (966 words)

	Polynomial Extensions of Finite Rings (Site not responding. Last check: 2007-10-01)
		for your base ring, the polynomial should not be cubic.
		As with all rings, the additive group is abelian.
		The presence of zerodivisors indicates that the modulus polynomial is not irreducible.
	www.hostsrv.com /webmaa/app1/MSP/webm1010/PolyExtensionRing.msp (83 words)

	Polynomial (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-10-01)
		In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions.
		In this article polynomials are written using a monomial basis (i.e.
		Polynomials With coefficients in R can be added by simply adding corresponding coefficients and multiplied using the distributive law and the rules :
	polynomial.iqnaut.net.cob-web.org:8888 (1747 words)

	Ring_Relations
		The set of consequences of a set of identities has the same arithmetic closure properties of an ideal – but, in addition, it is closed under the operation of substitution of polynomials for its variables.
		In other words, we want a computational test to determine if a polynomial identity is a consequence of an initial set of identities.
		Suppose cW is a term in a polynomial f.
	math.ucsd.edu /~jwavrik/web00/Ring_Relations.htm (1136 words)

	Twisted Polynomial Ring
		A twisted polynomial over a ring r is r adjoin two or more indeterminants, where some or all of the indeterminants do not commute with each other.
		If d is a division ring, the ring of twisted polynomials over d, in finitely many variables, is noetherian.
		This is an ascending chain, hence the polynomial ring is not left noetherian.
	www.mathreference.com /mod-acc,twist.html (713 words)

	Rings
		It is clear that polynomial multiplication does not, in general, have an inverse.
		One reason polynomials are of interest in signal processing is that polynomial multiplication is equivalent to convolution.
		In addition to the representing the arithmetic operations on sequences, polynomials can be used to represent a shift data.
	www.engineering.usu.edu /classes/ece/7670/lecture3/node2.html (251 words)

	Singular 2-0-5 Manual: Rings and orderings
		a quotient ring by an ideal of one of 1.
		Except for quotient rings, all of these rings are realized by choosing a coefficient field, ring variables, and an appropriate global or local monomial ordering on the ring variables.
		Objects of ring dependent types are local to a ring.
	web.mit.edu /singular_v2.0.5/distrib/Singular/2-0-5/html/sing_28.htm (176 words)

	GAP Manual: 19 Polynomials
		First of all, the polynomial (..., x_0 = 0, x_1 = 1, x_2 = 0,...) is commonly denoted by x and is called an indeterminate over R, (..., c_{-1}, c_0, c_1,...) is written as...
		The trivial polynomial and the zero of the base ring are always different.
		The other point which might be startling is that we require the supply of a base ring for a polynomial.
	www-groups.dcs.st-and.ac.uk /gap/Gap3/Manual3/C019S000.htm (720 words)

	18.5 Multivariate Polynomial Rings
		In this example, we construct a polynomial ring S in 3 variables over a polynomial ring in 2 variables over GF(9).
		Return True if this multivariate polynomial ring is a field, i.e., it is a ring in 0 generators over a field.
		Return dictionary of paris varname:var of the variables of this multivariate polynomial ring.
	modular.math.washington.edu /sage/doc/html/ref/module-sage.rings.multi-polynomial-ring.html (635 words)

	Laurent Polynomial -- from Wolfram MathWorld
		A Laurent polynomial with coefficients in the field
		A Laurent polynomial is an algebraic object in the sense that it is treated as a polynomial except that the indeterminant "
		is the group ring of the integers and the
	mathworld.wolfram.com /LaurentPolynomial.html (127 words)

	Non-commutative Elimination in Ore Algebras Proves Multivariate Identities
		Examples of skew polynomial rings are given in Table 1.
		One reason for studying these skew polynomial rings is that operations which can be performed in them need only be implemented once and then apply equally to linear differential equations, linear difference equations or their q-analogues.
		A left Ore ring is a ring such that for any non-zero elements a and b there exist non-zero U and V in the ring which satisfy Ua=Vb.
	algo.inria.fr /papers/html/ChSa96/ChSa96.html (6716 words)

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