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.
CITATION STYLE
Bhattacharyya, A. (2014). Polynomial decompositions in polynomial time. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8737 LNCS, pp. 125–136). Springer Verlag. https://doi.org/10.1007/978-3-662-44777-2_11
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