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Topic: Homogeneous polynomial

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	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/HomogeneousPolynomial.html (254 words)

	PlanetMath: homogeneous polynomial
		An equivalent definition is that all terms of the polynomial have the same degree (i.e.
		As an important example of homogeneous polynomials one can mention the symmetric polynomials.
		This is version 8 of homogeneous polynomial, born on 2003-01-04, modified 2005-02-26.
	planetmath.org /encyclopedia/HomogenousPolynomial.html (66 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.math.wisc.edu /help/magma/text403.html (778 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)

	Appendix to Part I: More details on NCCollectOnVariables
		For example, the polynomials in (5.2) and (5.3) are equal, but (5.3) is much nicer, since it makes it clear that the polynomial depends on A, B, X+Y and Z rather than just on A, B, X, Y and Z.
		Definition 9.1 Let p be a polynomial and V be a set of variables.
		When given a polynomial p and a set of variables V, NCCollect first writes the polynomial p as a sum of polynomials which are homogeneous in V.
	math.ucsd.edu /~ncalg/StrategyPaper/node53.html (329 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.
		In mathematics, a polynomial is an expression in which a finite number of constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power).
	www.bookrags.com /Polynomial (3406 words)

	LMI Papers
		Exploiting a suitable representation of homogeneous forms, a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one-parameter family of LMI optimization problems.
		A sufficient condition for the existence of a homogeneous polynomial Lyapunov function of given degree is formulated in terms of a Linear Matrix Inequalities (LMI) feasibility problem.
		For polynomial systems of degree m in n variables, a basic procedure is available if the kernel dimension does not exceed m+1, while an extended procedure can be applied if the kernel dimension is less than n(m-1)+2.
	www.ing.unisi.it /~garulli/opt_papers.html (1176 words)

	[No title]
		When each component of F is homogeneous of degree k, we will say that the polynomial map from X Rm to Y Rr is homogene* *ous of degree k.
		Representing elements of ssn(Sn) by polynomial maps is an old question [1] * which was affirmatively solved by Wood, in 1968, provided that n is odd (theorem 1 of * [3], see[4] as well for the complex sphere).
		In bot* h cases the polynomial* maps constructed are homogeneous; therefore the problem of repre* senting elements of ssn(Sn) by homogeneous* polynomial maps is solved now, since only ze* *ro and the odd topological degrees may be represented in this way when n is even [2].
	hopf.math.purdue.edu /Turiel/poly.txt (903 words)

	Projective Varieties
		The value of the polynomial is zero, or nonzero, regardless of the scaling factor l.
		In summary, the polynomials that vanish on specific regions in projective space are the homogeneous polynomials, where each term has degree d.
		A homogeneous ideal h in this ring is generated by homogeneous polynomials, and it has the happy property that any polynomial in h can be separated into homogeneous blocks, as described above, and these homogeneous subpolynomials are part of h.
	www.mathreference.com /ag-pv,intro.html (458 words)

	Algebraic Curves
		The notation as a non-homogeneous polynomial in 2 variables is convenient if we want to study the affine part of the curve (for example in the integral basis computation), but not if we are interested in the part of the curve on the line at infinity.
		This polynomial is a curve of degree 10 having a maximal number of cusps according to the Plucker formulas.
		The input of this procedure is a polynomial f in x and y, for which the singularity that we are interested in is located at x=0, y=0.
	www.math.fsu.edu /~hoeij/algcurves.html (2857 words)

	Homogeneous Coordinates
		Converting a degree n homogeneous polynomial in x, y, z and w back to the conventional form is exactly identical to the two-variable case.
		Given a homogeneous coordinate (x,y,w) of a point in the xy-plane, let us consider (x,y,w) to be a point in space whose coordinate values are x, y and w for the x-, y- and w- axes, respectively.
		Therefore, as a homogeneous point moves on a curve defined by homogeneous polynomial f(x,y,w)=0, its corresponding point moves in three-dimensional space, which, in turn, is projected to the plane w=1.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/geometry/homo-coor.html (1133 words)

	Advanced Factorization Techniques: The Number Field Sieve
		Returns a homogeneous degree d polynomial in two variables which is obtained by writing n in base m and adjusting coefficients to be between -m/2 and +m/2.
		The polynomials that are considered are base m polynomials having successive leading coefficients, and the corresponding values of m are chosen as to minimize the second to leading coefficient.
		Given a tuple that contains a univariate polynomial f, an integer m, a real number corresponding to the average log size of f for some optimal skewness value s, and optionally the real number s, this routine finds a translation of f such that the average log size is a local minimum.
	www.umich.edu /~gpcc/scs/magma/text544.htm (2191 words)

	Construction of Invariants of Specified Degree
		The homogeneous invariants in R of degree d form a vector space R_d over K. There are two ways of explicitly constructing homogeneous invariants in R of degree d: the Reynolds operator method and the linear algebra method.
		Construct a K-basis of the space R_d of the homogeneous invariants of degree d in the invariant ring R=K[V]^G of the group G over the field K as a sequence of polynomials.
		Construct k linearly independent homogeneous invariants of degree d in the invariant ring R=K[V]^G of the group G over the field K as a sequence of polynomials.
	www.umich.edu /~gpcc/scs/magma/text992.htm (625 words)

	The behavior of the principal distributions on the graph of a homogeneous polynomial, Naoya Ando
		The behavior of the principal distributions on the graph of a homogeneous polynomial, Naoya Ando
		The behavior of the principal distributions on the graph of a homogeneous polynomial
		In this paper, we shall study the behavior of the principal distributions on the graph of a homogeneous polynomial in two variables such that the set of its umbilical points is finite.
	projecteuclid.org /getRecord?id=euclid.tmj/1113247561 (215 words)

	Reduced norm in a central simple algebra (Site not responding. Last check: 2007-10-09)
		coefficient is a homogeneous polynomial of degree n
		(x) is a homogeneous polynomial of degree n-m in the coordinates of x.
		The reduced norm is a multiplicative map from A to k, given by a polynomial of degree n in the coefficients of x.
	www.math.harvard.edu /~elkies/M250.01/reduced.html (495 words)

	Symmetric polynomial - Wikipedia, the free encyclopedia
		In mathematics, a symmetric polynomial is a polynomial P(X
		Symmetric polynomials are important to linear algebra, representation theory, and Galois theory.
		The theorem may be proved for symmetric homogeneous polynomials by a double mathematical induction with respect to the number of variables n and, for fixed n, with respect to the degree of the homogeneous polynomial.
	en.wikipedia.org /wiki/Symmetric_polynomial (575 words)

	Solving equations
		Thue equations are Diophantine equations of the form f(x, y) = k, where k is some (integer) constant and f is a homogeneous polynomial in two variables.
		Given a polynomial of degree at least 2 over the integers, this function returns the `Thue object' corresponding to f; such objects are used in the functions for solving Thue equations, and print as the homogeneous version of f.
		Given a Thue object t and integers a, b, return the evaluation of the homogeneous polynomial f involved in t at (a, b), that is f(a, b).
	www.math.wisc.edu /help/magma/text454.html (515 words)

	Three Classes of Polynomial Systems
		The support A of a sparse polynomial p collects all exponents of those monomials whose coefficients are nonzero.
		The volume of a positive linear combination of polytopes is a homogeneous polynomial in the multiplication factors.
		The vanishing of a polynomial as in (5) expresses the condition that the r-plane X meets a given m-plane nontrivially.
	www.math.uic.edu /~jan/srvart/node3.html (424 words)

	Radical/Prime Ideals and Closed Sets
		A homogeneous ideal is just another ideal, so apply nullstellensatz, and rad(h) becomes the closure of h.
		Suppose p is a homogeneous polynomial in h, and not in j.
		The former is an intersection of homogeneous prime ideals, while the latter is a union of projective varieties.
	www.mathreference.com /ag-pv,rad.html (763 words)

	Generating Hilbert series
		The Hilbert polynomial of R is the numerical polynomial p(t) such that for n sufficiently large dim R_n = p(n).
		There is one extra twist: it is possible that there is not a single Hilbert polynomial p that determines the higher coefficients, but a sequence of polynomials p_0,..., p_(r - 1), and that the nth coefficient is given by p_i(n) where i is congruent to n modulo r.
		The Hilbert function corresponding to a univariate polynomial p, or a sequence Q of univariate polynomials, and a sequence of initial values V. The returned object is a function that can be evaluated at integers.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text1227.htm (1473 words)

	Homogeneous polynomial forms for simultaneous stabilizability of families of linear control systems: a tensor product ... (Site not responding. Last check: 2007-10-09)
		Homogeneous polynomial forms for simultaneous stabilizability of families of linear control systems: a tensor product approach
		The paper uses the formalism of tensor products in order to deal with the problem of simultaneous stabilizability of a family o f linear control systems by means of Lyapunov functions which are homogeneous polynomial forms.
		While the feedback synthesis seems to be nonconvex, the simultaneous stability by means of homogeneous polynomial forms of the uncontrollable modes yields (convex) necessary but not sufficient conditions for simultaneous stabilizability.
	www.sissa.it /~altafini/papers/hompol.html (107 words)

	ECCC Report TR04-070 and related Papers (Site not responding. Last check: 2007-10-09)
		Abstract: Let $p(x_1,...,x_n) =sum_{ (r_1,...,r_n) in I_{n,n} } a_{(r_1,...,r_n) } prod_{1 leq i leq n} x_{i}^{r_{i}}$ be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients.
		The support of such polynomial $p(x_1,...,x_n)$ is defined as $supp(p) = {(r_1,...,r_n) in I_{n,n} : a_{(r_1,...,r_n)} neq 0 }$.
		We study the following decision problems, which are far-reaching generalizations of the classical perfect matching problem : begin{itemize} item {bf Problem 1.} Consider a homogeneous polynomial $p(x_1,...,x_n)$ of degree $n$ in $n$ real variables with nonnegative integer coefficients given as a fl box (oracle).
	eccc.hpi-web.de /eccc-reports/2004/TR04-070/index.html (214 words)

	Gregorio Malajovich: Publications
		The bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the Newton polytopes (and underlying variances) are all identical.
		An important application of the method is factorizing polynomials or polynomial root-finding.
		Polynomials with real coefficients are also considered, and bounds for the expected number of real roots and for the condition number are given.
	www.labma.ufrj.br /~gregorio/papers.php (1984 words)

	Non-technical research description for Mike Knapp
		These are polynomials in several variables with the property that if you look at every term individually, then adding up the degrees of the variables in the term always gives you the same number.
		For example, one homogeneous polynomial of degree 5 would be
		Now, suppose that f(x) is a polynomial and p is a prime number (I'll do a couple of specific examples in a minute).
	www.evergreen.loyola.edu /~mpknapp/research.html (1966 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		= y; any polynomial equation in cartesian coordinates becomes homogeneous if a change into these coordinates is made.
		] An equation that can be rewritten into the form having zero on one side of the equal sign and a homogeneous function of all the variables on the other side.
		] A polynomial all of whose terms have the same total degree; equivalently it is a homogenous function of the variables involved.
	www.accessscience.com /Dictionary/H/H18/DictH18.html (1147 words)

	A.1 Numbers of polynomial terms (Site not responding. Last check: 2007-10-09)
		distinct monomials in an n-variable homogeneous polynomial of total degree r.
		The proposition holds for 1 variable (n = 1), because there is clearly 1 distinct monomial of each degree precisely r and hence at most d+1 distinct monomials in a polynomial of maximum degree d.
		Then a homogeneous polynomial of degree r in the n+1 variables X together with the single variable x has the form
	www.uni-koeln.de /REDUCE/3.6/doc/randpoly/node12.html (251 words)

	Mike Knapp - Papers and Preprints
		Abstract: In this paper we develop a bound on the number of variables required to guarantee that two diagonal homogeneous polynomials of different degrees k and n with coefficients in Q
		The purpose of this paper is to bound the primes for which exponential growth is required in the situation where the degrees are all different.
		This bound is larger than the Ax-Kochen bound, but we note that it applies for primes which are smaller than the largest of the degrees.
	www.evergreen.loyola.edu /~mpknapp/papers (1101 words)

	M-curves
		Any polynomial can be converted into a homogeneous polynomial.
		To convert a homogeneous polynomial to an affine polynomial, just reverse the above steps.
		Real polynomial: A real polynomial is a polynomial whose coefficients are all real.
	pages.prodigy.net /danesmith/mcurve/mcurve.html (889 words)

	AMCA: Identifiability of Homogeneous Polynomial Systems using the State Isomorphism Approach by Ralf Peeters (Site not responding. Last check: 2007-10-09)
		AMCA: Identifiability of Homogeneous Polynomial Systems using the State Isomorphism Approach by Ralf Peeters
		We will consider a class of homogeneous polynomial systems.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/l/t/17.htm (334 words)

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