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 $f(x_1, x_2) = x_1^2 + x_1x_2 + x_2^2$ 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 $\Phi$ of size $s$. Then, for any $r \leq d$ there is a circuit $\Psi$ of size $O(r^2s)$ that computes $H_0[f], H_1[f], \dots, H_r[f]$.
Proof: We will prove the theorem by constructing the circuit $\Psi$.