In algebra, a multivariate polynomial
![{\displaystyle f(x)=\sum _{\alpha }a_{\alpha }x^{\alpha }{\text{, where ))\alpha =(i_{1},\dots ,i_{r})\in \mathbb {N} ^{r}{\text{, and ))x^{\alpha }=x_{1}^{i_{1))\cdots x_{r}^{i_{r)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/865d3e7afd5f7637c3e80f365e40061170c70019)
is quasi-homogeneous or weighted homogeneous, if there exist r integers
, called weights of the variables, such that the sum
is the same for all nonzero terms of f. This sum w is the 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
![{\displaystyle f(\lambda ^{w_{1))x_{1},\ldots ,\lambda ^{w_{r))x_{r})=\lambda ^{w}f(x_{1},\ldots ,x_{r})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0224d8d8ba47d84aca93adad9181893daf1fd1f6)
for every
in any field containing the coefficients.
A polynomial
is quasi-homogeneous with weights
if and only if
![{\displaystyle f(y_{1}^{w_{1)),\ldots ,y_{n}^{w_{n)))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09dabc59fa5e77cf96d5d9f615b35b8c9390261)
is a homogeneous polynomial in the
. In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1.
A polynomial is quasi-homogeneous if and only if all the
belong to the same affine hyperplane. As the Newton polytope of the polynomial is the convex hull of the set
the quasi-homogeneous polynomials may also be defined as the polynomials that have a degenerate Newton polytope (here "degenerate" means "contained in some affine hyperplane").
Introduction
Consider the polynomial
, which is not homogeneous. However, if instead of considering
we use the pair
to test homogeneity, then
![{\displaystyle f(\lambda ^{3}x,\lambda y)=5(\lambda ^{3}x)^{3}(\lambda y)^{3}+(\lambda ^{3}x)(\lambda y)^{9}-2(\lambda y)^{12}=\lambda ^{12}f(x,y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abadd17d22b6613a2af5ea95caee7b1fa5102cc9)
We say that
is a quasi-homogeneous polynomial of type
(3,1), because its three pairs (i1, i2) of exponents (3,3), (1,9) and (0,12) all satisfy the linear equation
. In particular, this says that the Newton polytope of
lies in the affine space with equation
inside
.
The above equation is equivalent to this new one:
. Some authors[1] prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type
.
As noted above, a homogeneous polynomial
of degree d is just a quasi-homogeneous polynomial of type (1,1); in this case all its pairs of exponents will satisfy the equation
.
Definition
Let
be a polynomial in r variables
with coefficients in a commutative ring R. We express it as a finite sum
![{\displaystyle f(x)=\sum _{\alpha \in \mathbb {N} ^{r))a_{\alpha }x^{\alpha },\alpha =(i_{1},\ldots ,i_{r}),a_{\alpha }\in \mathbb {R} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cfcfab87b9ca2427d22d37bc4627370a0500e02)
We say that f is quasi-homogeneous of type
,
, if there exists some
such that
![{\displaystyle \langle \alpha ,\varphi \rangle =\sum _{k}^{r}i_{k}\varphi _{k}=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4378af3c352078254a54139baf6d2db5e3646cc3)
whenever
.