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., \(s^{{\mathrm{poly}}(d)}\) ). 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 \(\Sigma^{[2]}\Pi\Sigma\Pi^{[\mathsf{ind\hbox{-}deg} \; d]}\) circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.
