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Polynomial whose nonzero terms all have the same degree
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example
Homogeneous_polynomial
weight or the degree of the polynomial. The term quasi-homogeneous comes from the fact that a polynomial f is quasi-homogeneous if and only if f ( λ w 1
Quasi-homogeneous_polynomial
Function with a multiplicative scaling behaviour
to a kth-degree or kth-order homogeneous function. For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k. The above
Homogeneous_function
Mathematical concept in polynomial theory
and drawing of curves defined by a bivariate polynomial equation. The resultant of n homogeneous polynomials in n variables (also called multivariate resultant
Resultant
Function of the coefficients of a polynomial that gives information on its roots
quadratic form; and more generally, the discriminant of a form, of a homogeneous polynomial, or of a projective hypersurface (these three concepts are essentially
Discriminant
Mathematical function
elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Coordinate system used in projective geometry
^{k}f(x,y,z)=0.} A polynomial g ( x , y ) {\displaystyle g(x,y)} of degree k {\displaystyle k} can be turned into a homogeneous polynomial by replacing x
Homogeneous_coordinates
Type of symmetric polynomials in mathematics
elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible
Schur_polynomial
Expression in commutative algebra
complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression
Complete homogeneous symmetric polynomial
Complete_homogeneous_symmetric_polynomial
Type of mathematical expression
a polynomial is called homogeneous of degree n {\displaystyle n} if all of its non-zero terms have degree n {\displaystyle n} . The zero polynomial is
Polynomial
Polynomial equation whose integer solutions are sought
Diophantine equations. A homogeneous Diophantine equation is a Diophantine equation that is defined by a homogeneous polynomial. A typical such equation
Diophantine_equation
Tool in mathematical dimension theory
quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. The quotient by an ideal of a multivariate polynomial ring, filtered
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
and algebraic geometry, the multi-homogeneous Bézout theorem is a generalization to multi-homogeneous polynomials of Bézout's theorem, which counts the
Multi-homogeneous Bézout theorem
Multi-homogeneous_Bézout_theorem
Algebraic structure
especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally
Polynomial_ring
Curve defined as zeros of polynomials
set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables
Algebraic_curve
Algebraic variety in a projective space
P n {\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety
Projective_variety
projective Nullstellensatz states that, for any homogeneous ideal I that does not contain all polynomials of a certain degree (referred to as an irrelevant
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only
Polynomial_SOS
Number of intersection points of algebraic curves and hypersurfaces
defined by homogeneous polynomials in n + 1 indeterminates, then N is either infinite, or equals the product of the degrees of the polynomials. Moreover
Bézout's_theorem
Polynomial with all terms of degree two
mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4 x 2 + 2 x y
Quadratic_form
Mathematical object studied in the field of algebraic geometry
in k[x0, ..., xn] be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in Pn in homogeneous coordinates. However, because
Algebraic_variety
Relations between power sums and elementary symmetric functions
of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable
Newton's_identities
Concept of uniform or non-uniform in an object's composition or attributes
algebra, homogeneous polynomials have the same number of factors of a given kind.[citation needed] In the study of binary relations, a homogeneous relation
Homogeneity_and_heterogeneity
Mathematical concept
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The
Degree_of_a_polynomial
In mathematics, invariant of square matrices
multilinearity results from the fact that the Leibniz formula is a homogeneous polynomial of degree one in the entries of each column of the matrix; the fact
Determinant
Endofunctor on the category V of finite-dimensional vector spaces
theory (the calculus of functors). In particular, the category of homogeneous polynomial functors of degree n is equivalent to the category of finite-dimensional
Polynomial_functor
Manifold or algebraic variety of dimension n in a space of dimension n+1
by a homogeneous polynomial P ( x 0 , x 1 , … , x n ) {\displaystyle P(x_{0},x_{1},\ldots ,x_{n})} in n + 1 indeterminates. As usual, homogeneous polynomial
Hypersurface
Special mathematical functions defined on the surface of a sphere
"spherical harmonics" for these functions. The solid harmonics were homogeneous polynomial solutions R 3 → R {\displaystyle \mathbb {R} ^{3}\to \mathbb {R}
Spherical_harmonics
{q+n-3 \choose n-1}} such monomials. Thus, there exists a nonzero homogeneous polynomial P ( x 1 , x 2 , … , x n ) {\displaystyle P(x_{1},x_{2},\dots ,x_{n})}
Polynomial method in combinatorics
Polynomial_method_in_combinatorics
Polynomial with only one term
and multivariate polynomials. Explicitly, it is used to define the degree of a polynomial and the notion of homogeneous polynomial, as well as for graded
Monomial
Mathematical concept
x 1 , … , x n ] d {\displaystyle K[x_{1},\ldots ,x_{n}]_{d}} of homogeneous polynomial of degree d in n variables over the field K is the number of weak
Composition_(combinatorics)
Mathematical algorithm
recursive method to calculate the coefficients of the characteristic polynomial p A ( λ ) = det ( λ I n − A ) {\displaystyle p_{A}(\lambda )=\det(\lambda
Faddeev–LeVerrier_algorithm
Branch of mathematics
algorithm for solving systems of homogeneous polynomial equations with a computational complexity which is essentially polynomial in the expected number of solutions
Algebraic_geometry
Type of algebraic structure
{\displaystyle R_{i}} consisting of homogeneous polynomials of degree i. Let S be the set of all nonzero homogeneous elements in a graded integral domain
Graded_ring
K[X0, X1, X2, ..., XN] is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective
Homogeneous_coordinate_ring
Topics referred to by the same term
doctrine Concomitant (classical algebraic geometry), an invariant homogeneous polynomial in the coefficients of a form, a covariant variable, and a contravariant
Concomitant
Polynomial invariant under variable permutations
a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomials play
Symmetric_polynomial
Differential equation that is linear with respect to the unknown function
function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. The equation
Linear_differential_equation
Topics referred to by the same term
dictionary. Quantic may refer to: Quantic, an older name for a homogeneous polynomial. Quantic Dream, a video game developer studio Quantic (musician)
Quantic
Topics referred to by the same term
forms used in Chinese martial arts and sport wushu Algebraic form (homogeneous polynomial), which generalises quadratic forms to degrees 3 and more, also
Form
Generalization of the Jack polynomial
Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and
Jack_function
called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables
Quasi-algebraically closed field
Quasi-algebraically_closed_field
Expression of polynomials as sum of squares
restricted to homogeneous polynomials of even degree, since a polynomial of odd degree changes sign, and the homogenization of a polynomial takes only nonnegative
Hilbert's_seventeenth_problem
Mathematical formula for generating function
group G on a finite-dimensional vector space, that counts the homogeneous polynomials of a given total degree that are invariants for G. It is named
Molien's_formula
the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in R {\displaystyle \mathbb {R} } or C {\displaystyle
Bombieri_norm
power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients
Power sum symmetric polynomial
Power_sum_symmetric_polynomial
Topology on prime ideals and algebraic varieties
many representatives that yield different values in a polynomial; however, for homogeneous polynomials the condition of having zero or nonzero value on any
Zariski_topology
Homogeneous polynomial of degree 3
In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic
Cubic_form
are in the sum, the polynomial is said to be n-homogeneous. We define the space Pn as consisting of all n-homogeneous polynomials. The P1 is identical
Polynomially_reflexive_space
Class of polynomials related to Brahmagupta's identity
In algebra, Brahmagupta polynomials are a class of polynomials associated with the Brahmagupta matrix, which in turn is associated with Brahmagupta's identity
Brahmagupta_polynomials
Set of lines described by homogeneous polynomial equations
list of homogeneous polynomial equations. That is, a projective variety of lines. A linear line complex is defined by a list of degree-1 polynomials. A quadratic
Line_complex
+X_{n}^{2})^{m}p} is a sum of squares of homogeneous polynomials of degree m + 2 k {\displaystyle m+2k} . For polynomials of degree ≤ 1 {\displaystyle {}\leq
Positive_polynomial
Formula relating the norm and the inner product in an inner product space
generalizes the latter formula, replacing Q {\displaystyle Q} by a homogeneous polynomial of degree k {\displaystyle k} defined by Q ( v ) = B ( v , … , v
Polarization_identity
On the existence of zeros of homogeneous polynomials over the p-adic numbers
prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic numbers in at least d2 + 1 variables
Ax–Kochen_theorem
"Smallest" commutative algebra that contains a vector space
symmetric algebra of L (a polynomial ring) by the homogeneous ideal generated by the elements of M, which are homogeneous of degree one. One can also
Symmetric_algebra
Greatest common divisior of two polynomials Symmetric function Homogeneous polynomial Polynomial SOS (sum of squares) Polynomial family Quadratic function Cubic
List_of_polynomial_topics
Generalized alphabetical order
property of the degree reverse lexicographical order is that a homogeneous polynomial is a multiple of the least indeterminate if and only if its leading
Lexicographic_order
zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials. Zonal
Zonal_polynomial
binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms. It is a generalization of a Cunningham number
Binomial_number
Term in mathematics
m homogeneous polynomials: F i ( X 0 , ⋯ , X n ) , 1 ≤ i ≤ n − m , {\displaystyle F_{i}(X_{0},\cdots ,X_{n}),1\leq i\leq n-m,} in the homogeneous coordinates
Complete_intersection
Typically linear operator defined in terms of differentiation of functions
point x of X, the symbol σ P {\displaystyle \sigma _{P}} defines a homogeneous polynomial of degree k in T x ∗ X {\displaystyle T_{x}^{*}X} with values in
Differential_operator
Uniformity of a material or system at every point
James. "homogeneous." Encyclopedia of Mathematics. New York: Facts On File, Inc., 2005. Science Online. Facts On File, Inc. "A polynomial in several
Homogeneity_(physics)
Type of mathematical curve
defined by a homogeneous polynomial of degree 3 in three variables F ( X , Y , Z ) {\displaystyle F(X,Y,Z)} or by the corresponding polynomial in two variables
Plane_cubic_curve
Technique for expressing a polynomial in simpler fashion by using more variables
for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces
Polarization of an algebraic form
Polarization_of_an_algebraic_form
Generalizations of codimension-1 subvarieties of algebraic varieties
subvariety of a projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by
Divisor_(algebraic_geometry)
Swedish mathematician and concert pianist
exists C ( m , n ) > 0 {\displaystyle C(m,n)>0} such that for all homogeneous polynomials P {\displaystyle P} and Q {\displaystyle Q} of degrees m {\displaystyle
Per_Enflo
order polynomial is a polynomial studied in mathematics, in particular in algebraic graph theory and algebraic combinatorics. The order polynomial counts
Order_polynomial
by homogeneous symmetric functions of positive degree. The Schubert polynomial S w {\displaystyle {\mathfrak {S}}_{w}} is the unique homogeneous polynomial
Schubert_polynomial
German mathematician (1882–1935)
ask for the invariants of homogeneous polynomials A0xry0 + ... + Arx0yr of higher degree, which will be certain polynomials in the coefficients A0, .
Emmy_Noether
Algebraic structure
's. Any λ in S q ( V ) {\displaystyle S^{q}(V)} gives rise to a homogeneous polynomial function f of degree q: we just let f ( v ) = λ ( v , … , v ) .
Ring_of_polynomial_functions
Type of homogenous polynomial
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is of
Diagonal_form
On the number of spanning trees in a graph
from the original Laplacian matrix), above is then a homogeneous polynomial (the Kirchhoff polynomial) in the indeterminates corresponding to the edges of
Kirchhoff's_theorem
Integral transform in mathematics
ds} is the restriction to S n − 1 {\displaystyle S^{n-1}} of a homogeneous polynomial in α {\displaystyle \alpha } of degree k {\displaystyle k} . Indeed
Radon_transform
Type of equation with integer coefficients
f(x,y)=r,} where f {\displaystyle f} is an irreducible bivariate homogeneous polynomial of degree at least 3 over the rational numbers, and r {\displaystyle
Thue_equation
Type of integral domain
the variables Xi are given weights wi, and F(X1, ..., Xn) is a homogeneous polynomial of weight w. Then if c is coprime to w and R is a UFD and either
Unique_factorization_domain
Function that is invariant under all permutations of its variables
vector space V {\displaystyle V} is isomorphic to the space of homogeneous polynomials of degree k {\displaystyle k} on V . {\displaystyle V.} Symmetric
Symmetric_function
Unsolved problem in geometry
algebraic variety, that is, it is the zero set of a collection of homogeneous polynomials. Another way of phrasing the Hodge conjecture involves the idea
Hodge_conjecture
Ideal generated by differences of monomials
or projective algebraic variety defined by a toric prime ideal or a homogeneous toric ideal is an affine or projective toric variety. Miller, Ezra; Sturmfels
Toric_ideal
Algebraic structure
perfect. Finite fields are quasi-algebraically closed: every degree d homogeneous polynomial in n variables over a finite field with n > d > 0 {\displaystyle
Finite_field
Integral of the Gaussian function, equal to sqrt(π)
integral of the exponential of a homogeneous polynomial in n variables may depend only on SL(n)-invariants of the polynomial. One such invariant is the discriminant
Gaussian_integral
Generalization of vector bundles
on P n {\displaystyle \mathbb {P} ^{n}} . In particular, every homogeneous polynomial in x 0 , … , x n {\displaystyle x_{0},\ldots ,x_{n}} of degree j
Coherent_sheaf
In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables. Hilbert (1888) showed that a positive semi-definite ternary
Ternary_quartic
Method of representing a random variable
Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and Wiener chaos expansion, is a method for representing a random variable in terms
Polynomial_chaos
Plane algebraic curve
Equivalently, if the curve is determined in homogeneous coordinates by G(x, y, z) = 0, where G is a homogeneous polynomial, then the curve is circular if and only
Circular_algebraic_curve
Several equations of degree 1 to be solved simultaneously
Linear equation over a ring. For coefficients and solutions that are polynomials, see Gröbner basis. For finding the "best" integer solutions among many
System_of_linear_equations
of a polynomial which is convex but not SOS-convex was constructed by Amir Ali Ahmadi and Pablo Parrilo in 2009. The polynomial is a homogeneous polynomial
SOS-convexity
Relation between algebraic varieties and polynomial ideals
_{\mathbb {P} ^{n}}(S)} is the homogeneous ideal generated by homogeneous polynomials f that vanish on S. Now, for any homogeneous ideal I ⊆ R + {\displaystyle
Hilbert's_Nullstellensatz
Compact non-orientable two-dimensional manifold
restricts to a map whose domain is S2 and, since each component is a homogeneous polynomial of even degree, it takes the same values in R4 on each of any two
Real_projective_plane
Mathematical concept
one polynomial equation f ( x , y ) = 0 {\displaystyle f(x,y)=0} (or F ( x , y , z ) = 0 , {\displaystyle F(x,y,z)=0,} where F is a homogeneous polynomial
Plane_curve
Algebraic structure with addition, multiplication, and division
application of this is the Ax–Kochen theorem describing zeros of homogeneous polynomials in Qp. Tamely ramified extensions of both fields are in bijection
Field_(mathematics)
About polynomials in several variables
Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n {\displaystyle n} -dimensional
Jacobian_conjecture
Topics referred to by the same term
polynomial function of degree three Cubic equation, a polynomial equation (reducible to ax3 + bx2 + cx + d = 0) Cubic form, a homogeneous polynomial of
Cubic
Euclidean space without distance and angles
as a change of affine coordinates may map indeterminates on non-homogeneous polynomials. Affine spaces over topological fields, such as the real or the
Affine_space
Matrix of geometric progressions
0+1+2+\cdots +n={\frac {n(n+1)}{2}};} (that is, the determinant is a homogeneous polynomial of this degree). If, for i ≠ j {\displaystyle i\neq j} , one substitutes
Vandermonde_matrix
Quadratic homogeneous polynomial in two variables
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables q ( x , y ) = a x 2 + b x y + c y 2 , {\displaystyle q(x
Binary_quadratic_form
Distinguished element of a Lie algebra's center
have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order. Suppose that g {\displaystyle {\mathfrak
Casimir_element
In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective
Quaternary_cubic
Type of differential equation
constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being
Partial_differential_equation
Statement about cubic curves in the projective plane
out of P1, ..., P8 are co-conic, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) P1, ..., P8 (with multiplicity
Cayley–Bacharach_theorem
Solutions of the Laplace equation in spherical polar coordinates
regular solid harmonics correspond to harmonic homogeneous polynomials, i.e. homogeneous polynomials which are solutions to Laplace's equation. Racah's
Solid_harmonics
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HOMOGENEOUS POLYNOMIAL
HOMOGENEOUS POLYNOMIAL
HOMOGENEOUS POLYNOMIAL
HOMOGENEOUS POLYNOMIAL
HOMOGENEOUS POLYNOMIAL
HOMOGENEOUS POLYNOMIAL
HOMOGENEOUS POLYNOMIAL
HOMOGENEOUS POLYNOMIAL
HOMOGENEOUS POLYNOMIAL
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