Quasi-homogeneous polynomial

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A quasi-homogeneous polynomial is a polynomial which has a degenerate Newton polygon. This means that if

$f(x)=\sum _{\alpha }a_{\alpha }x^{\alpha }{\text{, where }}\alpha =(i_{1},\dots ,i_{r})\in {\mathbb {N}}^{r}{\text{, and }}x^{\alpha }=x_{1}^{{i_{1}}}\cdots x_{r}^{{i_{r}}}$

is a polynomial, then there r integers $w_{1},\ldots ,w_{r}$ , called weights of the variables such that the sum $w=w_{1}i_{1}+\cdots +w_{r}i_{r}$ is the same for all terms of f. This sum is called the weight or the degree of the polynomial. In other words, the convex hull of the set $\{\alpha |a_{\alpha }\neq 0\}$ lies entirely on an affine hyperplane.

The term quasi-homogeneous comes form the fact that a polynomial f is quasi-homogeneous if and only if

$f(\lambda ^{{w_{1}}}x_{1},\ldots ,\lambda ^{{w_{r}}}x_{r})=\lambda ^{w}f(x_{1},\ldots ,x_{r})$

for every $\lambda$ in the field of the coefficients. A homogeneous polynomial is quasi-homogeneous for all weights equal to 1.

Introduction

Consider the polynomial $f(x,y)=5x^{3}y^{3}+xy^{9}-2y^{{12}}$ . This one has no chance of being a homogeneous polynomial; however if instead of considering $f(\lambda x,\lambda y)$ we use the pair $(\lambda ^{3},\lambda )$ to test homogeneity, then

$f(\lambda ^{3}x,\lambda y)=5(\lambda ^{3}x)^{3}(\lambda y)^{3}+(\lambda ^{3}x)(\lambda y)^{9}-2(\lambda y)^{{12}}=\lambda ^{{12}}f(x,y).\,$

We say that $f(x,y)$ 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 $3i_{1}+1i_{2}=12$ . In particular, this says that the Newton polygon of $f(x,y)$ lies in the affine space with equation $3x+y=12$ inside ${\mathbb {R}}^{2}$ .

The above equation is equivalent to this new one: ${\tfrac {1}{4}}x+{\tfrac {1}{12}}y=1$ . Some authors^[1] prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type ( ${\tfrac {1}{4}},{\tfrac {1}{12}}$ ).

As noted above, a homogeneous polynomial $g(x,y)$ 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 $1i_{1}+1i_{2}=d$ .

Definition

Let $f(x)$ be a polynomial in r variables $x=x_{1}\ldots x_{r}$ with coefficients in a commutative ring R. We express it as a finite sum

$f(x)=\sum _{{\alpha \in {\mathbb {N}}^{r}}}a_{\alpha }x^{\alpha },\alpha =(i_{1},\ldots ,i_{r}),a_{\alpha }\in {\mathbb {R}}.$

We say that f is quasi-homogeneous of type $\varphi =(\varphi _{1},\ldots ,\varphi _{r})$ , $\varphi _{i}\in {\mathbb {N}}$ if there exists some $a\in {\mathbb {R}}$ such that

$\langle \alpha ,\varphi \rangle =\sum _{k}^{r}i_{k}\varphi _{k}=a,$

whenever $a_{\alpha }\neq 0$ .

References

↑ J. Steenbrink (1977). Compositio Mathematica, tome 34, n° 2. Noordhoff International Publishing. p. 211 (Available on-line at Numdam)

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