We show how to efficiently count exactly the number of solutions of a system of n polynomials in n variables over certain local fields L, for a new class of polynomials systems. The fields we handle include the reals and the p-adic rationals. The polynomial systems amenable to our methods are made up of certain A-discriminant chambers, and our algorithms are the first to attain polynomial-time in this setting. We also discuss connections to Baker's refinement of the abc-Conjecture, Smale's 17th Problem, and tropical geometry. The results presented are, in various combinations, joint with Martin Avendano, Philippe Pebay, Korben Rusek, and David C. Thompson.