Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding

COUNTING TROPICALLY DEGENERATE VALUATIONS AND P-ADIC APPROACHES TO THE HARDNESS OF THE PERMANENT

Abstract. The Shub-Smale τ-Conjecture is a hitherto unproven statement (on integer roots of polynomials) whose truth implies both a variant of P=NP (for the BSS model over C) and the hardness of the permanent. We give alternative conjectures, some clearly easier to prove, whose truth still implies the hardness of the permanent. Along the way, we discuss new upper bounds on the number of p-adic valuations of roots of certain sparse polynomial systems, culminating in a connection between quantitative p-adic geometry and complexity theory. Dedicated to Mike Shub, on his 70 th birthday. 1