Ensemble learning

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

$f(x)=\sum _{\alpha }{a}_{\alpha }{x}^{\alpha }\text{, where }\alpha =(i_1,\dots ,i_r)\in {\mathbb{\mathbb{N}}}^r\text{, and }{x}^{\alpha }=x_1^{i_1}\cdots x_r^{i_r}$

is a polynomial, then there r integers $w_1,\dots ,w_r$, 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 $\{\alpha |a_{\alpha }\ne 0\}$ 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

$f({\lambda }^{w_1}{x}_1,\dots ,{\lambda }^{w_r}{x}_r)={\lambda }^{w}f(x_1,\dots ,x_r)$

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 $f(x,y)=5x^3{y}^3+x{y}^9-2y^{12}$. 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: $\frac{1}{4}x+\frac{1}{12}y=1$. Some authors prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type ($\frac{1}{4},\frac{1}{12}$).

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 $x=x_1\dots x_r$ with coefficients in a commutative ring R. We express it as a finite sum

$f(x)=\sum _{\alpha \in {\mathbb{\mathbb{N}}}^r}{a}_{\alpha }{x}^{\alpha },\alpha =(i_1,\dots ,i_r),a_{\alpha }\in \mathbb{ℝ}.$

We say that f is quasi-homogeneous of type $\phi =({\phi }_1,\dots ,{\phi }_r)$, ${\phi }_i\in \mathbb{\mathbb{N}}$ if there exists some $a\in \mathbb{ℝ}$ such that

$⟨\alpha ,\phi ⟩=\sum _k^r{i}_k{\phi }_k=a,$

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