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

Suppose $A=\{a_1,\ldots,a_{n+2}\}\subset\mathbb{Z}^n$ 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 $F=(f_1,\ldots,f_n)$ is an $n\times n$ 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 an algorithm that, for any fixed $n$, counts exactly the number of roots of $F$ in the positive orthant in time polynomial in $\log(dH)$. (The fastest previous algorithms had exponential dependence on $\log d$, already for $n=2$.) The use of Diophantine approximation over number fields, to identify the underlying discriminant chamber containing $F$, plays a key role. Our underlying estimates are also useful for certifying numerical solutions of certain sparse polynomial systems.