Fewnomial systems with many roots, and an Adelic Tau Conjecture
Kaitlyn Phillipson, J. Maurice Rojas · Contemporary mathematics - American Mathematical Society · 2013
Consider a system F F of n n polynomials in n n variables, with a total of n + k n+k distinct exponent vectors, over any local field L L . We discuss conjecturally tight bounds on the maximal number of nondegenerate roots F F can have over L L , with all coordinates having fixed phase, as a function of n n , k k , and L 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 τ \tau -Conjecture and the P \mathbf {P} vs. N P \mathbf {NP} Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.