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

TL;DR

This paper introduces a polynomial-time algorithm for counting the exact number of real roots of certain polynomial systems supported on circuits, with complexity depending logarithmically on coefficient and degree bounds.

Contribution

It presents the first algorithm capable of counting real roots efficiently for systems supported on circuits with fixed dimension.

Findings

Algorithm runs in polynomial time in log(dH) for fixed n

Exact real root count for systems supported on circuits

Applicable to systems with integer coefficients and bounded degrees

Abstract

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 the first algorithm that, for any fixed $n$, counts exactly the number of real roots of $F$ in in time polynomial in $\log(dH)$.