New Explicit Root Counts for p-adic Circuit Systems

Fewnomial theory began by establishing (finite) upper bounds on the number of real non-degenerate roots of any n × n system of real polynomials, solely as a function of n and the number of distinct exponent vectors in the underlying monomial term expansions. Khovanski and Sevastyanov proved the first explicit bounds around 1980 [5, 6], and then an analogue over the p -adic rationals, ℚ _p , (with p ∈ ℕ prime) was found by Denef and van den Dries around 1988 [3, Sec. 3.4]. However, explicit bounds over ℚ _p are known only when n = 1 ([3, Sec. 3.5], [8, Thm. 2 & pg. 279]), or when n ≥ 2 and the underlying zero sets satisfy a very restrictive condition: tropical genericity (with respect to p) (see [11] and below). We give the first explicit upper bounds over ℚ _p (for p ≥ n + 1) for a broad class of tropically non -generic systems with n + 2 exponent vectors, where no bounds had been available before: See Theorem 1.5 below. Let us first review the simplest systems over R and ℚ _p .