On Solving Sparse Polynomial Factorization Related Problems - Bisht, Pranav; Volkovich, Ilya (Computational complexity, 2025)
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 ] Π Σ Π [ ind - deg d ] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.