  Consider a system F of n polynomials in n variables, with a total of n+k
distinct exponent vectors, over any local field L. We discuss conjecturally
tight bounds on the maximal number of non-degenerate roots F can have over L,
with all coordinates having fixed phase, as a function of n, k, and L only. In
particular, we give new explicit systems with number of roots approaching the
best known upper bounds. We also briefly review the background behind such
bounds, and their application, including connections to computational number
theory and variants of the Shub-Smale tau-Conjecture and the P vs. NP Problem.
One of our key tools is the construction of combinatorially constrained
tropical varieties with maximally many intersections.
