Counting Tropically Degenerate Valuations and p-adic Approaches to the Hardness of the Permanent

The Shub-Smale Tau Conjecture is a hitherto unproven statement (on integer roots of polynomials) whose truth implies both a variant of $P\mathrm{\ne }NP$ (for the BSS model over C) and the hardness of the permanent. We give alternative conjectures, some potentially 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.