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...theorem]], which counts the number of isolated common zeros of a set of [[homogeneous polynomial]]s. This generalization is due to [[Igor Shafarevich]].<ref>{{ci ...ns is finite, their number is bounded by the product of the degrees of the polynomials. Moreover, if the number of solutions [[point at infinity|at infinity]] is ...

is '''quasi-homogeneous''' or '''weighted homogeneous''', if there exist ''r'' integers <math>w_1, \ldots, w_r</math>, called ''' ...omogeneous'' comes from the fact that a polynomial {{math|''f''}} is quasi-homogeneous if and only if ...

...orics]], the '''Stanley symmetric functions''' are a family of [[symmetric polynomials|symmetric functions]] introduced by {{harvs|txt|authorlink=Richard P. Stanl ...]]. When a Stanley symmetric function is expanded in the basis of [[Schur polynomials|Schur functions]], the coefficients are all [[non-negative]] [[integer]]s. ...

...eous linear three-term recurrence relation''' ('''TTRR''', the qualifiers "homogeneous linear" are usually taken for granted)<ref>Gi, Segura, Temme (2007), Chapte [[Orthogonal polynomials]] ''P''_''n'' all have a TTRR with respect to degree ''n'', ...

...ry theory]] (the [[calculus of functors]]). In particular, the category of homogeneous polynomial functors of degree ''n'' is equivalent to the [[category of repr ...alued polynomial <math>F(\lambda_1 f_1 + \cdots + \lambda_r f_r)</math> is homogeneous of degree ''n''. ...

...'(''x'') where ''S'' is a matrix (possibly rectangular) which entries are polynomials in ''x''.<ref>{{Cite journal|last1=Helton|first1=J. William|last2=Nie|first ...he equivalence of algebraic conditions for convexity and quasiconvexity of polynomials |date=2010|pages=3343–3348|doi=10.1109/CDC.2010.5717510|hdl=1721.1/74151|is ...

Let ''K''_''n'' be polynomials over a [[ring (mathematics)|ring]] ''A'' in indeterminates ''p''<sub>1</sub ...m''&nbsp;>&nbsp;''j''. This is symmetric in the ''β''_''i'' and homogeneous of weight ''j'': so can be expressed as a polynomial ''K''_''j''( ...

== Homogeneous polynomial ideals== ...d by '''k'''[''V'']. It is a naturally [[graded algebra]] by the degree of polynomials. ...

...unction of the coefficients, which takes the value zero if and only if the polynomials have a common non-trivial zero over some field containing the coefficients. ...which is homogeneous in some variables, the resultant ''eliminates'' these homogeneous variables by providing an equation in the other variables, which has, as so ...

...ebra of symmetric functions]] and that of [[Symmetric polynomial|symmetric polynomials]]. It is essentially basic substitution of variables, but allows for a cha *If <math>f</math> is a homogeneous symmetric function of degree <math>d</math>, then ...

...a/Scanned/36-1/suryanarayan.pdf |access-date=1 December 2023}}</ref> These polynomials have several interesting properties and have found applications in [[Tessel === Brahmagupta polynomials === ...

Multilinear polynomials can be understood as a [[multilinear map]] (specifically, a [[multilinear f ...es, where the weights are the [[Lagrange polynomial|Lagrange interpolation polynomials]]. These weights also constitute a set of [[generalized barycentric coordin ...

{{For|Jacobi polynomials of several variables|Heckman–Opdam polynomials}} ...ematics]], '''Jacobi polynomials''' (occasionally called '''hypergeometric polynomials''') <math>P_n^{(\alpha,\beta)}(x)</math> ...

In mathematics, '''Schubert polynomials''' are generalizations of [[Schur polynomials]] that represent cohomology classes of [[Schubert cycle]]s in [[Generalized {{harvtxt|Lascoux|1995}} described the history of Schubert polynomials. ...

...under all permutations of their variables|the generalization of symmetric polynomials to infinitely many variables (in algebraic combinatorics)|ring of symmetric ...tor space]] <math>V</math> is [[isomorphic]] to the space of [[homogeneous polynomials]] of degree <math>k</math> on <math>V.</math> Symmetric functions should no ...

...]] and [[Power sum symmetric polynomial|power sums]] homogeneous symmetric polynomials in many variables. Its name comes from the operation called [[plethysm]], d ...on]] for many well studied sequences of [[Integer|integers]], [[Polynomial|polynomials]] or power series, such as the number of integer [[List of partition topics ...

Given generic homogeneous polynomials <math> g_1,g_2,\ldots,g_k\in \mathbb{C}[x_1,x_2,\ldots,x_n]</math> of degre ...

...hism (and, actually, the algebras are not isomorphic, since the algebra of polynomials is infinite while that of polynomial functions is finite). Any λ in <math>S^q(V)</math> gives rise to a [[homogeneous polynomial]] function ''f'' of [[Degree of a polynomial|degree]] ''q'': we ...

{{About|positive polynomials and positivstellensatz-like theorems|the Krivine–Stengle Positivstellensatz For certain sets <math>S</math>, there exist algebraic descriptions of all polynomials that are positive (resp. non-negative) on <math>S</math>. Such a descripti ...

where ''h''_''r'' is a [[complete homogeneous symmetric polynomial]] and the sum is over all partitions &lambda; obtained ...I. G. | author1-link=Ian G. Macdonald | title=Symmetric functions and Hall polynomials | url=http://www.oup.com/uk/catalogue/?ci=9780198504504 | archive-url=https ...