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Homogeneous polynomial

In mathematics, a homogeneous polynomial of degree d in n variables is a polynomial that consists solely of monomials each of total degree d, expressible as a linear combination of such terms like x_1^{i_1} x_2^{i_2} \cdots x_n^{i_n} where i_1 + i_2 + \cdots + i_n = d.^[1] This structure ensures that scaling the variables by a factor t scales the polynomial by t^d, i.e., f(tx_1, \dots, tx_n) = t^d f(x_1, \dots, x_n), a property that distinguishes homogeneous polynomials from general ones.^[2] Key properties include the finite-dimensionality of the vector space of homogeneous polynomials of fixed degree d in n variables, with dimension given by the binomial coefficient \binom{d + n - 1}{n - 1}, representing the number of distinct monomials.^[1] Euler's theorem provides a fundamental relation: for a homogeneous polynomial f of degree d over the reals, the dot product of the variables with the gradient equals d times the polynomial, \sum_{i=1}^n x_i \frac{\partial f}{\partial x_i} = d f(x).^[3] These polynomials play a central role in algebraic geometry, where the zero sets of homogeneous ideals define projective varieties in projective space \mathbb{P}^n, enabling the study of geometric objects invariant under scaling.^[4] Beyond geometry, homogeneous polynomials arise in diverse applications, such as modeling level sets that form curves or surfaces with arithmetic properties, like conic sections or elliptic curves, and in optimization problems across sciences including robotics and biology.^[1]^[5] They also underpin the study of symmetric polynomials and invariants in group theory, facilitating computations in arithmetic complexity and differential equations.^[6]

Fundamentals

Definition

In mathematics, a homogeneous polynomial in n variables x_1, \dots, x_n over a field (such as the real or complex numbers) is a polynomial p(x_1, \dots, x_n) of degree d where every monomial term has total degree exactly d. Formally, it can be expressed as

p(x_1, \dots, x_n) = \sum_{i_1 + \dots + i_n = d} c_{i_1 \dots i_n} x_1^{i_1} \cdots x_n^{i_n},

with coefficients c_{i_1 \dots i_n} in the field, and the sum taken over all non-negative integers i_1, \dots, i_n satisfying the degree condition.^[7]^[1] This uniformity distinguishes homogeneous polynomials from general polynomials, which may include terms of varying degrees, allowing the former to exhibit specific scaling behaviors under variable rescaling.^[7] In the univariate case (n=1), a homogeneous polynomial simplifies to p(x) = c x^d for some constant c and degree d \geq 0, making it a monomial whose properties, such as roots and factorization, follow directly from those of powers.^[1]^[8] The zero polynomial, with all coefficients zero, is conventionally considered homogeneous of every degree d, as it vacuously satisfies the degree condition for any d and ensures consistency in algebraic structures like graded rings.^[9]

Examples

Homogeneous polynomials appear in both univariate and multivariate settings, providing simple yet illustrative cases of the concept. In the univariate case, a homogeneous polynomial of degree d consists solely of a single term, or monomial, of that degree, such as x^2 (degree 2) or $3x^5 (degree 5), where scaling the variable by a factor \lambda yields \lambda^d times the original polynomial.^[10] The zero polynomial, with all coefficients zero, is conventionally regarded as homogeneous of every degree d \geq 0, as it satisfies the homogeneity condition trivially for any such d.^[11] In the bivariate case, terms of the same total degree combine to form homogeneous polynomials. For instance, x^2 + 2xy + y^2 is homogeneous of degree 2, representing the square of a linear form. Another example is x^3 - 3xy^2, which is homogeneous of degree 3 and corresponds to the real part of the cube of the complex variable z = x + iy, i.e., \operatorname{Re}(z^3).^[12]^[13] Trivariate examples further demonstrate the structure in higher dimensions. The polynomial x^2 + y^2 + z^2 is a homogeneous quadratic form of degree 2, often used to define spheres or norms in vector spaces. Similarly, xy - z^2 is homogeneous of degree 2 and arises in the study of conic sections in projective geometry, such as the equation defining a hyperbola in homogeneous coordinates.^[7]^[14] Polynomials that mix terms of different degrees are not homogeneous; for example, x^2 + y combines a degree-2 term with a degree-1 term and thus fails the homogeneity condition. Geometrically, homogeneous polynomials without constant terms represent functions invariant under scaling of variables, capturing projective or radial behaviors in spaces like cones or algebraic varieties.^[12]

Algebraic Properties

Scaling and Homogeneity

A homogeneous polynomial p of degree d in n variables satisfies the scaling property: for any scalar \lambda, p(\lambda x_1, \dots, \lambda x_n) = \lambda^d p(x_1, \dots, x_n).^[7] This property arises directly from the structure of the polynomial, as it is a sum of monomials each of total degree d. To see this, consider a general monomial term c x_1^{a_1} \cdots x_n^{a_n} where \sum_{i=1}^n a_i = d; substituting scaled variables yields c (\lambda x_1)^{a_1} \cdots (\lambda x_n)^{a_n} = c \lambda^d x_1^{a_1} \cdots x_n^{a_n} = \lambda^d (c x_1^{a_1} \cdots x_n^{a_n}). Summing over all such terms gives the full scaling relation for p.^[2] This scaling implies that a homogeneous polynomial remains unchanged under variable scaling only if d = 0, in which case p is a nonzero constant (up to the field's scalars), as \lambda^0 = 1 for \lambda \neq 0. For d > 0, the polynomial scales by a nontrivial power of \lambda, reflecting its degree.^[2] The scaling property extends multiplicatively to compositions of homogeneous polynomials. Specifically, if p is homogeneous of degree d and q: \mathbb{R}^n \to \mathbb{R}^m has each component homogeneous of degree e, then the composition p \circ q is homogeneous of degree d e. To verify, note that q(\lambda \mathbf{x}) = \lambda^e q(\mathbf{x}), so p(q(\lambda \mathbf{x})) = p(\lambda^e q(\mathbf{x})) = (\lambda^e)^d p(q(\mathbf{x})) = \lambda^{d e} p(q(\mathbf{x})).^[15] To confirm that a polynomial is homogeneous of degree d, one can verify the scaling property on a set of sample points or basis monomials, ensuring each term scales by exactly \lambda^d and no lower- or higher-degree terms appear.^[2]

Euler's Homogeneous Function Theorem

Euler's homogeneous function theorem provides a fundamental relation between a homogeneous polynomial and its partial derivatives. For a homogeneous polynomial p(x_1, \dots, x_n) of degree d, the theorem states that

\sum_{i=1}^n x_i \frac{\partial p}{\partial x_i} = d \, p(x_1, \dots, x_n).

This identity arises directly from the scaling property of homogeneous polynomials, where p(\lambda x_1, \dots, \lambda x_n) = \lambda^d p(x_1, \dots, x_n) for any \lambda > 0.^[2]^[16] To derive this, consider the scaling equation and apply logarithmic differentiation with respect to \lambda. Differentiate both sides of p(\lambda \mathbf{x}) = \lambda^d p(\mathbf{x}) using the chain rule: the left side yields \sum_{i=1}^n \frac{\partial p}{\partial (\lambda x_i)} x_i, and the right side gives d \lambda^{d-1} p(\mathbf{x}). Evaluating at \lambda = 1 produces the theorem's identity. Alternatively, define g(\lambda) = p(\lambda \mathbf{x}) - \lambda^d p(\mathbf{x}), note that g(\lambda) = 0 for all \lambda > 0, so g'(\lambda) = 0, and apply the chain rule at \lambda = 1 to obtain the same result.^[3]^[2] Special cases illustrate the theorem's behavior for low degrees. For degree d = 0, the polynomial is constant (nonzero only if independent of variables), all partial derivatives vanish, and the identity simplifies to \sum_{i=1}^n x_i \frac{\partial p}{\partial x_i} = 0. For degree d = 1, the polynomial is linear, the partial derivatives are constants (the coefficients), and the sum equals the polynomial itself: \sum_{i=1}^n x_i \frac{\partial p}{\partial x_i} = p.^[17]^[16] Although the theorem extends to general homogeneous functions (continuously differentiable and satisfying the scaling condition), its application here is restricted to polynomials, where homogeneity implies all monomials share the same total degree d. This ensures the partial derivatives are themselves homogeneous polynomials of degree d-1.^[3]^[2] Computationally, the theorem offers a practical method to verify or determine the degree of a suspected homogeneous polynomial. Compute the partial derivatives, form the weighted sum \sum_{i=1}^n x_i \frac{\partial p}{\partial x_i}, and equate it to d p; solving for d (typically by comparing coefficients or evaluating at a test point) yields the degree, provided the identity holds.^[16]^[2]

Construction Methods

Homogenization

Homogenization is the process of transforming a non-homogeneous polynomial f(x_1, \dots, x_n) into a homogeneous polynomial by introducing an auxiliary variable t, ensuring all terms have the same total degree.^[18] For a polynomial f(x_1, \dots, x_n) = \sum_k f_k(x_1, \dots, x_n), where each f_k is the homogeneous component of degree k, the homogenization F(x_1, \dots, x_n, t) of total degree d is given by

F(x_1, \dots, x_n, t) = \sum_k t^{d - k} f_k(x_1, \dots, x_n).

This construction multiplies each lower-degree term by an appropriate power of t to equalize degrees.^[18] The choice of d is typically the maximum degree of f, ensuring F is homogeneous of that degree without unnecessary higher powers of t.^[18] For instance, consider f(x, y) = x^2 + x y + y, which has terms of degrees 2, 2, and 1, respectively; its homogenization with d = 2 is F(x, y, t) = x^2 + x y + y t.^[19] This homogenization is unique up to the selection of d, with the standard convention using the maximum degree to preserve the polynomial's structure minimally.^[18] The original polynomial recovers from the homogenized form by substituting t = 1, yielding f(x_1, \dots, x_n) = F(x_1, \dots, x_n, 1).^[18]

Dehomogenization

Dehomogenization is the process of obtaining a polynomial in the original variables from a homogeneous polynomial in an extended set of variables, typically by substituting the auxiliary variable with 1. For a homogeneous polynomial F(x_1, \dots, x_n, t) of degree d, the dehomogenized polynomial is defined as f(x_1, \dots, x_n) = F(x_1, \dots, x_n, 1). This operation yields a polynomial f whose terms may have degrees ranging from 0 to d, as the substitution mixes the homogeneous components.^[18] In the algebraic setting, dehomogenization can be performed with respect to any of the variables in a homogeneous polynomial F(x_0, \dots, x_n), by setting that variable to 1 and expressing the result in the remaining variables. For instance, dehomogenization with respect to x_i gives f(x_0, \dots, \hat{x_i}, \dots, x_n) = F(x_0, \dots, x_{i-1}, 1, x_{i+1}, \dots, x_n). This flexibility allows recovery of different affine forms from the same homogeneous polynomial, though the focus remains on the resulting non-homogeneous expressions.^[20] A concrete example illustrates the process: consider the homogeneous polynomial F(x, y, t) = x^2 + x y + y t of degree 2. Dehomogenizing by setting t = 1 produces f(x, y) = x^2 + x y + y, which is a non-homogeneous polynomial of mixed degrees. This reversal aligns with the homogenization procedure, serving as its inverse when applied to a polynomial that originated as non-homogeneous, provided the degree d is known to ensure uniqueness.^[18] However, dehomogenization has inherent limitations in the algebraic context. It does not preserve all structural information from the homogeneous form, such as scaling properties across the full variable set, and recovering the original homogeneous polynomial requires prior knowledge of the degree to avoid ambiguity in re-homogenization. Without this, multiple homogeneous extensions of degree greater than or equal to that of f could map to the same dehomogenized result.^[20]

Multivariable Extensions

Homogeneous Components in Multivariate Polynomials

In multivariate polynomials over a field k, any polynomial p \in k[x_1, \dots, x_n] of total degree d admits a unique decomposition as a direct sum p = \sum_{k=0}^d p_k, where each p_k is a homogeneous polynomial of exact degree k. This decomposition arises from the \mathbb{Z}_{\geq 0}-grading on the polynomial ring, where the graded piece of degree k is the k-dimensional vector space spanned by all monomials of total degree k, and the components p_k are the unique projections onto these pieces.^[21]^[22] The homogeneous component p_k can be theoretically extracted using differentiation with respect to a scaling parameter \lambda. Specifically, p(\lambda \mathbf{x}) = \sum_{k=0}^d \lambda^k p_k(\mathbf{x}), so p_k(\mathbf{x}) = \frac{1}{k!} \left. \frac{d^k}{d \lambda^k} p(\lambda \mathbf{x}) \right|_{\lambda=0}. In practice, the decomposition is obtained by collecting all monomials with the same total degree into separate sums.^[21] This grading equips the polynomial ring with an orthogonal structure under the direct sum, ensuring the components are unique and the ring is \mathbb{N}-graded, which facilitates many algebraic constructions and analyses.^[21] For example, consider p(x,y) = x^2 + xy + y^3. The decomposition is p = p_2 + p_3, where p_2(x,y) = x^2 + xy, and p_3(x,y) = y^3.

Relation to Symmetric Polynomials

Symmetric homogeneous polynomials are a special class of homogeneous polynomials that remain invariant under any permutation of their variables. These polynomials form the ring of invariants of the polynomial ring under the action of the symmetric group S_n. Prominent examples include the elementary symmetric polynomials e_k(x_1, \dots, x_n), which are the sums of all distinct products of k variables and are homogeneous of degree k, and the power sum polynomials p_k(x_1, \dots, x_n) = \sum_{i=1}^n x_i^k, also homogeneous of degree k.^[23]^[24] The ring of symmetric polynomials in n variables over the integers is freely generated as a polynomial ring by the first n elementary symmetric polynomials e_1, \dots, e_n, according to the fundamental theorem of symmetric polynomials; equivalently, it can be generated by the complete homogeneous symmetric polynomials h_1, \dots, h_n, where h_k sums all monomials of degree k in non-decreasing order of variables. This generating structure arises from the generating function for symmetric polynomials, such as the product \prod_{i=1}^n (1 + x_i t) = \sum_{k=0}^n e_k t^k, which encodes the elementary symmetric polynomials.^[24]^[25]^[23] In the context of invariant theory, the elementary symmetric polynomials provide a Hilbert basis for the ring of symmetric polynomials, meaning every symmetric polynomial can be expressed as a polynomial in these generators with integer coefficients, and the ring is finitely generated as an algebra over \mathbb{Z}. This ties directly to Hilbert's finiteness theorem for invariants under finite group actions, where the symmetric group S_n yields polynomial invariants generated by the e_k. For instance, the homogeneous polynomial x^2 + y^2 + z^2 of degree 2 can be expressed in the symmetric basis as (x + y + z)^2 - 2(xy + yz + zx) = e_1^2 - 2 e_2.^[24]^[25] Schur polynomials represent another fundamental family of homogeneous symmetric polynomials, defined combinatorially via semistandard Young tableaux of a given shape \lambda: s_\lambda(x_1, \dots, x_n) is the sum over all such tableaux of the monomial given by the content of the tableau, and it is homogeneous of degree |\lambda|. They form an orthonormal basis for the ring of symmetric polynomials with respect to the Hall scalar product and play a key role in representations of the symmetric group.^[26]^[23]

Applications

In Algebraic Geometry

In algebraic geometry, homogeneous polynomials play a fundamental role in defining hypersurfaces within projective space \mathbb{P}^n. A hypersurface in \mathbb{P}^n is the zero locus of an irreducible homogeneous polynomial f(x_0, \dots, x_n) of degree d, denoted V(f) = \{ \in \mathbb{P}^n \mid f(x) = 0 \}, where $$ represents equivalence classes of points under scaling by nonzero scalars in the base field.^[4] This construction is invariant under scaling because if f(x) = 0, then f(\lambda x) = \lambda^d f(x) = 0 for \lambda \neq 0, ensuring the zero set projects well to projective space.^[27] The degree of the hypersurface equals the degree of the polynomial, capturing essential geometric invariants such as genus and intersection behavior.^[4] Homogeneous ideals, generated by homogeneous polynomials, are graded ideals in the polynomial ring k[x_0, \dots, x_n] and are central to the study of projective schemes. The zero set of such an ideal I in \mathbb{P}^n forms a projective variety V(I), which is the projectivization of the affine cone C(V(I)) over it.^[28] These ideals ensure that the coordinate ring k[V(I)] = k[x_0, \dots, x_n]/I(V(I)) is graded, facilitating the use of sheaf theory and cohomology on projective schemes.^[28] For instance, the ideal sheaf of a hypersurface defined by a homogeneous polynomial corresponds to the twisting sheaf \mathcal{O}(-d) on \mathbb{P}^n.^[4] Bézout's theorem quantifies the intersections of homogeneous hypersurfaces, stating that two curves of degrees m and n in \mathbb{P}^2 intersect in exactly mn points, counted with multiplicity, assuming the base field is algebraically closed.^[4] This extends to higher dimensions for complete intersections of hypersurfaces, where the intersection multiplicity at a point is the dimension of the localized quotient ring.^[29] A classic example is quadratic forms, which define conics in \mathbb{P}^2; for instance, the equation x^2 + y^2 - z^2 = 0 describes a smooth conic, projectively equivalent to the standard circle, with self-intersection properties governed by Bézout's theorem.^[4] The connection to affine geometry arises through dehomogenization, where setting one variable (e.g., x_0 = 1) yields affine charts covering \mathbb{P}^n, transforming homogeneous equations into affine polynomials that describe local pieces of the variety.^[27]

In Invariant Theory

In invariant theory, a homogeneous polynomial p of degree d on a vector space V is called an invariant under the action of a linear algebraic group G, such as \mathrm{GL}(n) or \mathrm{SL}(n), if p(g \cdot v) = p(v) for all g \in G and v \in V.^[30] This condition ensures that p remains unchanged under linear transformations induced by the group, making it a symmetry-preserving quantity often restricted to specific degrees. The space of such invariants forms a subring of the polynomial ring on V, graded by degree, and plays a central role in classifying orbits under group actions.^[30] A classical example arises with quadratic forms, where the discriminant serves as a fundamental homogeneous invariant. For a quadratic form represented by a symmetric matrix A, the discriminant is the square-free part of \det(A), which remains invariant under congruence transformations A \mapsto M^T A M for M \in \mathrm{GL}(n), as the determinant transforms by \det(M)^2 \det(A), preserving the class modulo squares.^[31] This invariance classifies quadratic forms up to equivalence, providing essential algebraic structure without altering the form's geometric properties.^[31] Hilbert's finiteness theorem asserts that the ring of invariants \mathbb{C}[V]^G for a reductive group G acting linearly on V is finitely generated as a \mathbb{C}-algebra, and moreover, it admits a homogeneous basis consisting of a finite set of invariant polynomials.^[32] Proven in 1890 using the basis theorem for ideals, this result extends earlier work on binary forms and guarantees that all invariants can be expressed algebraically in terms of these generators, facilitating computational approaches in invariant theory.^[32] The Reynolds operator provides a key tool for constructing invariants, defined as a linear projection R_G: \mathbb{C}[V] \to \mathbb{C}[V]^G that fixes invariants and is itself [G](/page/Group)-equivariant, often realized by averaging over the group for finite [G](/page/Group) as R_G(p) = \frac{1}{|G|} \sum_{g \in [G](/page/Group)} g \cdot p.^[33] For reductive groups, an analogous operator exists, preserving degrees of homogeneous components and enabling the decomposition of polynomials into invariant and non-invariant parts; it is multiplicative on invariants and crucial in proving finite generation via Hilbert's theorem.^[33] In the context of binary forms—homogeneous polynomials in two variables under \mathrm{SL}(2)—invariants such as the resultant exemplify degree-specific symmetries. The resultant of two binary forms of degrees m and n is a bihomogeneous invariant of bidegree (n, m) that vanishes precisely when the forms share a common root, serving as a discriminant-like measure for their intersection properties.^[34] This application underscores the role of invariants in solving systems and classifying forms, with the full ring generated by a finite set including the discriminant and Hessian for low degrees.^[34]

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