On Solving Sparse Polynomial Factorization Related Problems

In a recent result of Bhargava, Saraf and Volkovich [FOCS’18; JACM’20], the first factor sparsity bound for constant individual degree polynomials was shown. In particular, it was shown that any factor of a polynomial with at most s terms and individual degree bounded by d can itself have at most s O ( d 2 log n ) terms. It is conjectured, though, that the “true” sparsity bound should be polynomial (i.e. s poly( d ) ). In this paper we provide supporting evidence for this conjecture by presenting polynomial-time algorithms for several problems that would be implied by a polynomial-size sparsity bound. In particular, we give efficient (deterministic) algorithms for identity testing of Σ [2] ΠΣΠ [ ind - deg d ] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.