Polynomial Decompositions in Polynomial Time

Abstract

Fix a prime p. Given a positive integer k, a vector of positive integers Δ = (Δ₁, Δ₂, …, Δ _k ) and a function $\Gamma: \mathbb{F}_p^k \to \mathbb{F}_p$, we say that a function $P: \mathbb{F}_p^n \to \mathbb{F}_p$ is (k,Δ,Γ)-structured if there exist polynomials $P_1, P_2, \dots, P_k:\mathbb{F}_p^n \to \mathbb{F}_p$ with each deg(P _i ) ≤ Δ _i such that for all $x \in \mathbb{F}_p^n$,

$$P(x) = \Gamma(P_1(x), P_2(x), ..., P_k(x)).$$

For instance, an n-variate polynomial over the field $\mathbb{F}_p$ of total degree d factors nontrivially exactly when it is (2, (d-1,d-1), prod)-structured where prod(a,b) = a·b.

We show that if p > d, then for any fixed k, Δ, Γ, we can decide whether a given polynomial P(x ₁, x ₂, …, x _n ) of degree d is (k, Δ, Γ)-structured and if so, find a witnessing decomposition. The algorithm takes poly(n) time. Our approach is based on higher-order Fourier analysis.