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In mathematics, a homogeneous polynomial is a polynomial whose terms are monomials all having the same total degree; or are elements of the same dimension. For example, x5 + 2x3y2 + 9x1y4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. An algebraic form, or simply form, is another name for a homogeneous polynomial. A homogeneous polynomial of degree 2 is a quadratic form, and may be simply represented as a symmetric matrix. The theory of algebraic forms is very extensive, and has numerous applications all over mathematics and theoretical physics. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
In mathematics, a monomial (or mononomial) is a particular kind of polynomial, having just one term. ...
This article is about the term degree as used in mathematics. ...
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...
In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...
Symmetric tensors
Homogeneous polynomials over a vector space may be constructed directly from symmetric tensors, and vice versa. For vector spaces over the real or complex numbers, these the set of homogeneous polynomials and symmetric tensors are in fact isomorphic. This relationship is often expressed as follows. In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
Let X and Y be vector spaces, and let T the multi-linear map or symmetric tensor In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
 Define the diagonal operator Δ as  The homogeneous polynomial  of degree n associated with T is simply  , so that  Written this way, it is clear that a homogeneous polynomial is a homogeneous function of degree n. That is, for a scalar a, one has In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ...
 which follows immediately from the multi-linearity of the tensor.
Conversely, given a homogeneous polynomial P, one may construct the corresponding symmetric tensor  by means of the polarization formula: In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. ...
 Let  denote the space of symmetric tensors of rank n, and let  denote the space of homogeneous polynomials of degree n. If the vector spaces X and Y are over the reals or the complex numbers (or more generally, over a field of characteristic zero), then these two spaces are isomorphic, with the mappings given by hat and check: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
 and 
Algebraic forms in general Algebraic form, or simply form, is another term for homogeneous polynomial. These then generalise from quadratic forms to degrees 3 and more, and have in the past also been known as quantics. To specify a type of form, one has to give its degree of a form, and number of variables n. A form is over some given field K, if it maps from Kn to K, where n is the number of variables of the form. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
A form over some field K in n variables represents 0 if there exists an element
- (x1,...,xn)
in Kn such that at least one of the - xi (i=1,...,n)
is not equal to zero.
Basic properties The number of different homogeneous monomials of degree M in N variables is 
The Taylor series for a homogeneous polynomial P expanded at point x may be written as As the degree of the Taylor series rises, it approaches the correct function. ...
 Another useful identity is 
History Algebraic forms played an important role in nineteenth century mathematics. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
The two obvious areas where these would be applied were projective geometry, and number theory (then less in fashion). The geometric use was connected with invariant theory. There is a general linear group acting on any given space of quantics, and this group action is potentially a fruitful way to classify certain algebraic varieties (for example cubic hypersurfaces in a given number of variables). Projective geometry is a non-metrical form of geometry. ...
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...
In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ...
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ...
In mathematics, a symmetry group describes all symmetries of objects. ...
This article is about algebraic varieties. ...
In more modern language the spaces of quantics are identified with the symmetric tensors of a given degree constructed from the tensor powers of a vector space V of dimension m. (This is straightforward provided we work over a field of characteristic zero). That is, we take the n-fold tensor product of V with itself and take the subspace invariant under the symmetric group as it permutes factors. This definition specifies how GL(V) will act. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...
In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...
It would be a possible direct method in algebraic geometry, to study the orbits of this action. More precisely the orbits for the action on the projective space formed from the vector space of symmetric tensors. The construction of invariants would be the theory of the co-ordinate ring of the 'space' of orbits, assuming that 'space' exists. No direct answer to that was given, until the geometric invariant theory of David Mumford; so the invariants of quantics were studied directly. Heroic calculations were performed, in an era leading up to the work of David Hilbert on the qualitative theory. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
In mathematics, groups are often used to describe symmetries of objects. ...
This article does not cite its references or sources. ...
In mathematics, geometric invariant theory in algebraic geometry is a (technically complex) development building on nineteenth century invariant theory. ...
David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. ...
David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...
For algebraic forms with integer coefficients, generalisations of the classical results on quadratic forms to forms of higher degree motivated much investigation.
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