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

Abstract

Suppose $A=\{a_1,\dots ,a_{n+2}\}\subset {\mathbb{Z}}^n$ 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 $F=(f_1,\dots ,f_n)$ 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 $\log (dH)$.

29 pages, 1 figure, accepted for presentation at MEGA (Effective Methods in Algebraic Geometry) 2021. You can see a recording of my talk at MEGA 2021 (June 9, 2021) at this YouTube link: https://www.youtube.com/watch?v=KKKmTctxbs4

References in corpus (2)

• Bounded-degree factors of lacunary multivariate polynomials
• Optimal Descartes' Rule of Signs for Circuits