Counting Real Roots in Polynomial-Time for Systems Supported on Circuits

Abstract

Suppose A = {a 1 , . . . , a n+2 } ⊂ Z n has cardinality n + 2, with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = (f 1 , . . . , f n ) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in log(dH).We prove Theorem 1.1 in Section 5.1, based mainly on Algorithms 4.1 and 4.3 from Section 4. A central ingredient is diophantine approximation.The genericity condition for counting roots in (R * ) n and R n + , when t = n + 2, is the nonsingularity of a collection of sub-matrices of [c i,j ], depending solely on the support of F , and

Trinomials and Deterministic Complexity Limits for Real Solving

“…Recently, it was shown that one can count the real roots of circuit systems in deterministic polynomial-time, for any fixed n [26]: The proof reduced to proving the simplification of Conjecture 1.9 where one only asks for the number of real roots of g. This provides some slight evidence for Conjecture 1.9. More to the point, the framework from [26] reveals that proving Conjecture 1.9 would be the next step toward polynomial-time real-solving for circuit systems for n > 1. Such speed-ups are currently known only for binomial systems so far [23], since real-solving for arbitrary n×n systems still has exponential-time worst-case complexity when n is fixed (see, e.g., [25]).…”

“…where the exponent vectors of all the f i are contained in a set A ⊂ Z n of cardinality n + 2, with A not lying in any affine hyperplane. Such an A is called a circuit (the terminology coming from combinatorics, instead of complexity theory), and such systems have been studied from the point of view of real solving and fewnomial theory since 2003 (if not earlier): See, e.g., [17,4,5,26]. In particular, it has been known at least since [4] that solving such systems over R reduces mainly to finding the real roots of univariate rational functions of the form…”