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# Polynomial Decompositions in Polynomial Time

Bhattacharyya, Arnab (2014) Polynomial Decompositions in Polynomial Time. In: 22nd Annual European Symposium on Algorithms (ESA) held as part of ALGO Meeting Location, SEP 08-10, 2014, Wroclaw, POLAND , pp. 125-136.

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Official URL: http://www.cs.cornell.edu/~kozen/papers/poly.pdf

## Abstract

Fix a prime p. Given a positive integer k, a vector of positive integers Delta = (Delta(1), Delta(2), ... , Delta(k)) and a function Gamma : F-p(k) -> F-p, we say that a function P : F-p(n) -> F-p is (k, Delta, Gamma)-structured if there exist polynomials P-1, P-2, ..., P-k : F-p(n) -> F-p with each deg(P-i) <= Delta(i) such that for all x is an element of F-p(n), P(x) = Gamma(P-1(x), P-2(x), ..., P-k(x)). For instance, an n-variate polynomial over the field Fp 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, Delta, Gamma, we can decide whether a given polynomial P(x(1), x(2), ..., x(n)) of degree d is (k, Delta, Gamma)-structured and if so, find a witnessing decomposition. The algorithm takes poly(n) time. Our approach is based on higher-order Fourier analysis.

Item Type: Conference Proceedings Publisher Copyright for this article belongs to the SPRINGER-VERLAG BERLIN, HEIDELBERGER PLATZ 3, D-14197 BERLIN, GERMANY Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 12 Jan 2015 07:02 12 Jan 2015 07:02 http://eprints.iisc.ernet.in/id/eprint/50612

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