Lecture 4 - Basic Structural Results

In this lecture, we will discuss certain structural results that are fundamental to the study of algebraic complexity theory.

Homogenization

We say that a polynomial $f(x_1,\dots ,x_n)$ is homogeneous if all the monomials in $f$ have the same degree. For example, the polynomial $f(x_1,x_2)=x_1^2+x_1{x}_2+x_2^2$ is homogeneous of degree $2$. From now on, we will use the term form to refer to homogeneous polynomials.

Given any non-homogeneous polynomial $f(x_1,\dots ,x_n)$, we can decompose it into a sum of its homogeneous components (by simply collecting all the terms of same degree). For each degree $k$, we denote by $H_k[f]$ the homogeneous component of $f$ of degree $k$. Then, we can write $f$ as $$f(x_1,\dots ,x_n)=\sum _{k=0}^d{H}_k[f],$$ where $d$ is the degree of $f$.

For example, the polynomial $f(x,y)=x^3+2xy+y^2$ has homogeneous components $H_0[f]=H_1[f]=0$, $H_2[f]=2xy+y^2$, and $H_3[f]=x^3$.

Homogenization is a process that takes a circuit computing a (possibly non-homogeneous) polynomial $f(x_1,\dots ,x_n)$ and produces a circuit computing the homogeneous components of $f$. We will see that we can perform homogenization in a way that does not increase the size of the circuit by much.

Theorem 1 (Homogenization): Let $f(x_1,\dots ,x_n)$ be a polynomial of degree $d$ that can be computed by a circuit $\mathrm{\Phi }$ of size $s$. Then, for any $r\le d$ there is a circuit $\mathrm{\Psi }$ of size $O(r^{2}s)$ that computes $H_0[f],H_1[f],\dots ,H_r[f]$.

Proof: We will prove the theorem by constructing the circuit $\mathrm{\Psi }$.

Division Elimination

References