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 \(x \in \mathbb{F}_p^n\),
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.
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Bhattacharyya, A. (2014). Polynomial Decompositions in Polynomial Time. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_11
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