Homogeneous polynomial
Encyclopedia
In mathematics
, a homogeneous polynomial is a polynomial
whose monomial
s with nonzero coefficients all have the
same total degree
. For example,
is a homogeneous polynomial
of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial
is not homogeneous, because the sum of exponents does not match from term to term. An algebraic form, or simply form, is another name for a homogeneous polynomial. The theory of algebraic forms is very extensive, and has numerous applications all over mathematics and theoretical physics.
A homogeneous polynomial of degree 0 is simply a scalar, while a homogeneous polynomial of degree 1 is a linear functional
, also known as a covector. A homogeneous polynomial of degree 2 is a quadratic form
, and, away from 2 (in a ring where 2 is invertible), may be identified with a symmetric 2-tensor, simply represented as a symmetric matrix, though at 2 there is a difference between quadratic forms and symmetric forms, notably for integral quadratic forms. Higher degree homogeneous forms may likewise be identified with symmetric k-tensors so long as
is invertible.
s, and these can often be chosen to be isomorphism
s, inverse to each other; see quadratic form: symmetric forms for discussion for quadratic forms.
Most significantly, over the real or complex numbers symmetric tensors are identical to homogeneous polynomials, but over the integers they are distinct concepts, as discussed at integral quadratic form.
In general, over a field
of characteristic zero symmetric tensors and homogeneous polynomials are naturally identified, and more generally if
is invertible,Properly, the polarization identity requires 
to be invertible, but repeated factors of 2 can be eliminated, 2 is already included in 
for 
and for 
these are both simply linear functionals. then symmetric d-tensors can be identified with homogeneous polynomials of degree d.
The distinction is that symmetric d-tensors
are a subspace
of the set of all d-tensors
the invariant
s under the symmetric group
– 
while homogeneous polynomials 
are a quotient space
of the d-tensors, the coinvariants under the symmetric group
this can be done for all grades of tensors, yielding:
There is thus a natural inclusion of symmetric tensors in tensors, then quotienting to polynomials, yielding
there is further a symmetrization map from polynomials to symmetric tensors, 
but in general neither of these maps need be onto or one-to-one. For example, not every integral quadratic form arises from a symmetric 2-form: 
does not, but 
does.
s; then homogeneous polynomials in X of degree d, taking values in Y, and symmetric d-tensors from X to Y are defined and related as follows.
Let T be a multi-linear map (tensor), which need not be symmetric:
Define the diagonal morphism
as
The homogeneous polynomial
of degree d associated with T is simply 
, so that
Written this way, it is clear that a homogeneous polynomial is a homogeneous function
of degree d. That is, for a scalar a, one has
which follows immediately from the multi-linearity of the tensor.
Conversely, given a homogeneous polynomial
, one may construct the corresponding symmetric tensor 
by means of the polarization formula
or "symmetrization map":
Let
denote the space of symmetric tensors of rank d, and let 
denote the space of homogeneous polynomials of degree d. If the vector spaces X and Y are over the reals or the complex numbers, more generally, over a field
of characteristic zero, most generally if
is invertible, then these two spaces are isomorphic, with the mappings given by hat and check:
and
K, if it maps from Kn 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
in Kn such that at least one of the
is not equal to zero.
The Taylor series
for a homogeneous polynomial P expanded at point x may be written as
Another useful identity is
can be homogenized
by introducing an additional variable
and defining
where
is the degree
of
.
For example,
.
A homogenized polynomial can be dehomogenized by setting the additional variable
.
.
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).
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 the symmetric 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 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.
For algebraic forms with integer coefficients, generalisations of the classical results on quadratic forms to forms of higher degree motivated much investigation.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a homogeneous polynomial is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
whose monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
s with nonzero coefficients all have the
same total degree
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
. For example,
is a homogeneous polynomialof degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial
is not homogeneous, because the sum of exponents does not match from term to term. An algebraic form, or simply form, is another name for a homogeneous polynomial. The theory of algebraic forms is very extensive, and has numerous applications all over mathematics and theoretical physics.A homogeneous polynomial of degree 0 is simply a scalar, while a homogeneous polynomial of degree 1 is a linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
, also known as a covector. A homogeneous polynomial of degree 2 is a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
, and, away from 2 (in a ring where 2 is invertible), may be identified with a symmetric 2-tensor, simply represented as a symmetric matrix, though at 2 there is a difference between quadratic forms and symmetric forms, notably for integral quadratic forms. Higher degree homogeneous forms may likewise be identified with symmetric k-tensors so long as
is invertible.Symmetric tensors
Homogeneous polynomials are closely related to symmetric tensors, and are often confused with them. In general they are different concepts, but in many important cases they can be identified as identical: there is a natural map from symmetric tensors to homogeneous polynomials, and from homogeneous polynomials to symmetric polynomialSymmetric polynomial
In mathematics, a symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial...
s, and these can often be chosen to be isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s, inverse to each other; see quadratic form: symmetric forms for discussion for quadratic forms.
Most significantly, over the real or complex numbers symmetric tensors are identical to homogeneous polynomials, but over the integers they are distinct concepts, as discussed at integral quadratic form.
In general, over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic zero symmetric tensors and homogeneous polynomials are naturally identified, and more generally if
is invertible,Properly, the polarization identity requires to be invertible, but repeated factors of 2 can be eliminated, 2 is already included in for and for these are both simply linear functionals. then symmetric d-tensors can be identified with homogeneous polynomials of degree d.The distinction is that symmetric d-tensors
are a subspaceSubspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
of the set of all d-tensors
the invariantInvariant
Invariant and invariance may have several meanings, among which are:- Computer science :* Invariant , an Expression whose value doesn't change during program execution* A type in overriding that is neither covariant nor contravariant...
s under the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
– while homogeneous polynomials are a quotient spaceQuotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of the d-tensors, the coinvariants under the symmetric group
this can be done for all grades of tensors, yielding:There is thus a natural inclusion of symmetric tensors in tensors, then quotienting to polynomials, yielding
there is further a symmetrization map from polynomials to symmetric tensors, but in general neither of these maps need be onto or one-to-one. For example, not every integral quadratic form arises from a symmetric 2-form: does not, but does.Details
Let X and Y be vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s; then homogeneous polynomials in X of degree d, taking values in Y, and symmetric d-tensors from X to Y are defined and related as follows.
Let T be a multi-linear map (tensor), which need not be symmetric:
Define the diagonal morphism
asThe homogeneous polynomial
of degree d associated with T is simply , so thatWritten this way, it is clear that a homogeneous polynomial is a homogeneous function
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
of degree d. That is, for a scalar a, one has
which follows immediately from the multi-linearity of the tensor.
Conversely, given a homogeneous polynomial
, one may construct the corresponding symmetric tensor by means of the polarization formulaPolarization of an algebraic form
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables...
or "symmetrization map":
Let
denote the space of symmetric tensors of rank d, and let denote the space of homogeneous polynomials of degree d. If the vector spaces X and Y are over the reals or the complex numbers, more generally, over a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic zero, most generally if
is invertible, then these two spaces are isomorphic, with the mappings given by hat and check: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 fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K, if it maps from Kn 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
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 isThe Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
for a homogeneous polynomial P expanded at point x may be written as
Another useful identity is
Homogenization
A non-homogeneous polynomial can be homogenizedby introducing an additional variable
and definingwhere
is the degreeDegree (mathematics)
In mathematics, there are several meanings of degree depending on the subject.- Unit of angle :A degree , usually denoted by ° , is a measurement of a plane angle, representing 1⁄360 of a turn...
of
.For example,
.A homogenized polynomial can be dehomogenized by setting the additional variable
.History
Algebraic forms played an important role in nineteenth century mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
.
The two obvious areas where these would be applied were projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
(then less in fashion). The geometric use was connected with invariant theory
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
. There is a general linear group
General linear group
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. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
acting on any given space of quantics, and this group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
is potentially a fruitful way to classify certain algebraic varieties (for example cubic hypersurfaces in a given number of variables).
In more modern language the spaces of quantics are identified with the symmetric tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
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
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
zero). That is, we take the n-fold tensor product of V with itself and take the subspace invariant under the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
as it permutes factors. This definition specifies how GL(V) will act.
It would be a possible direct method in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, to study the orbits of this action. More precisely the orbits for the action on the projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
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
Geometric invariant theory
In mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces...
of David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
; so the invariants of quantics were studied directly. Heroic calculations were performed, in an era leading up to the work of David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
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.
See also
- diagonal form
- graded algebraGraded algebraIn mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
- Homogeneous functionHomogeneous functionIn mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
- multilinear form
- multilinear map
- polarization of an algebraic formPolarization of an algebraic formIn mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables...
- Schur polynomialSchur polynomialIn mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of...
- Symbol of a differential operatorSymbol of a differential operatorIn mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this...