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 Γ: Fpk → Fp, we say that a function P: Fpn → Fp is (k,Δ,Γ)-structured if there exist polynomials with each deg(P i )≤Δ;bi e such that for all x ε F pn P(x)= Γ(p1(x), p2(x), ⋯, pk(x)). For instance, an n-variate polynomial over the field 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;gtd, then for any fixed k, Δ, Γ, we can decide whether a given polynomial P(x 1 x2, ⋯, xnb n e) 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. © 2014 Springer-Verlag Berlin Heidelberg.