- Homogeneous polynomial
In

mathematics , a**homogeneous polynomial**is apolynomial whose terms aremonomial s all having the same total degree; or are elements of the samedimension . For example, $x^5\; +\; 2\; x^3\; y^2\; +\; 9\; x^1\; y^4$ 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 aquadratic form , and may be simply represented as asymmetric matrix . The theory of algebraic forms is very extensive, and has numerous applications all over mathematics and theoretical physics.**ymmetric tensors**Homogeneous polynomials over a vector space may be constructed directly from

symmetric tensor s, and vice versa. For vector spaces over the real or complex numbers, the set of homogeneous polynomials and symmetric tensors are in factisomorphic . This relationship is often expressed as follows.Let "X" and "Y" be

vector space s, and let "T" be the multi-linear map or symmetric tensor:$egin\{matrix\}T:\; underbrace\{X\; imes\; X\; imes\; cdots\; imes\; X\}\; o\; Y\backslash \; n\; \backslash end\{matrix\}$

Define the

diagonal operator $Delta$ as:$egin\{matrix\}Delta:\; X\; o\; X^n\; \backslash \; x\; mapsto\; (x,x,dots,x)\; \backslash end\{matrix\}$

The homogeneous polynomial $widehat\{T\}$ of degree "n" associated with "T" is simply $widehat\{T\}\; =\; T\; circ\; Delta$, so that

:$widehat\{T\}(x)\; =\; (T\; circ\; Delta)\; (x)\; =\; T(x,x,ldots,x)$

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:$widehat\{T\}(ax)\; =\; a^n\; widehat\{T\}(x)$

which follows immediately from the multi-linearity of the tensor.

Conversely, given a homogeneous polynomial $P$, one may construct the corresponding symmetric tensor $check\{P\}$ by means of the polarization formula:

:$check\{P\}(x\_1,x\_2,cdots\; x\_n)\; =\; frac\{1\}\{2^n\; n!\}\; sum\_\{varepsilon\_i=pm\; 1\; atop\; 1le\; ile\; n\}varepsilon\_1varepsilon\_2cdotsvarepsilon\_n\; Pleft(sum\_\{i=1\}\; varepsilon\_i\; x\_i\; ight)$

Let $mathcal\{L\}(X^n,Y)$ denote the space of symmetric tensors of rank "n", and let $mathcal\{P\}(X,Y)$ 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::$widehat\{;\}:\; mathcal\{L\}(X^n,Y)\; o\; mathcal\{P\}(X,Y)$

and

:$check\{;\}:\; mathcal\{P\}(X,Y)\; o\; mathcal\{L\}(X^n,Y)$

**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 "K"^{"n"}to "K", where "n" is the number of variables of the form.A form over some field "K" in "n" variables "represents 0" if there exists an element

:("x"

_{"1"},...,"x"_{"n"})in "K"

^{"n"}such that at least one of the:"x"

_{"i"}("i"=1,...,"n")is not equal to zero.

**Basic properties**The number of different homogeneous monomials of degree M in N variables is $frac\{(M+N-1)!\}\{M!(N-1)!\}$

The

Taylor series for a homogeneous polynomial "P" expanded at point "x" may be written as :$egin\{matrix\}P(x+y)=\; sum\_\{j=0\}^n\; \{n\; choose\; j\}\; check\{P\}\; (underbrace\{x,x,dots\; ,x\}\; underbrace\{y,y,dots\; ,y\}\; ).\; \backslash \; j\; n-j\backslash end\{matrix\}$Another useful identity is :$egin\{matrix\}P(x)-P(y)=\; sum\_\{j=0\}^\{n-1\}\; \{n\; choose\; j\}\; check\{P\}\; (underbrace\{y,y,dots\; ,y\}\; underbrace\{(x-y),(x-y),dots\; ,(x-y)\}\; ).\; \backslash \; j\; n-j\backslash end\{matrix\}$

**History**Algebraic forms played an important role in nineteenth century

mathematics .The two obvious areas where these would be applied were

projective geometry , andnumber theory (then less in fashion). The geometric use was connected withinvariant theory . There is ageneral linear group acting on any given space of quantics, and thisgroup action is potentially a fruitful way to classify certainalgebraic varieties (for examplecubic hypersurface s in a given number of variables).In more modern language the spaces of quantics are identified with the symmetric

tensor s 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 thesymmetric group as it permutes factors. This definition specifies how "GL(V)" will act.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 theprojective 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 thegeometric invariant theory ofDavid Mumford ; so the invariants of quantics were studied directly. Heroic calculations were performed, in an era leading up to the work ofDavid Hilbert on the qualitative theory.For algebraic forms with integer coefficients, generalisations of the classical results on quadratic forms to forms of higher degree motivated much investigation.

**ee also***

diagonal form

*graded algebra

*multilinear form

*multilinear map

*Schur polynomial

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2010.*

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