Counting solutions of a polynomial system locally and exactly

Counting Solutions of a Polynomial System Locally and Exactly

We propose a symbolic-numeric algorithm to count the number of solutions of a polynomial system within a local region. More specifically, given a zero-dimensional system f1 = · · · = fn = 0, with fi ∈ C[x1, . . . , xn], and a polydisc ∆ ⊂ C, our method aims to certify the existence of k solutions (counted with multiplicity) within the polydisc. In case of success, it yields the correct result u...

Counting Stable Solutions of Sparse Polynomial Systems

Counting Solutions to Polynomial Systems via Reductions

This paper provides both positive and negative results for counting solutions to systems of polynomial equations over a finite field. The general idea is to try to reduce the problem to counting solutions to a single polynomial, where the task is easier. In both cases, simple methods are utilized that we expect will have wider applicability (far beyond algebra). First, we give an efficient dete...

A Deterministic Polynomial-Time Approximation Scheme for Counting Knapsack Solutions

Given n elements with nonnegative integer weights w1, . . . , wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error 1±ε in time polynomial in n and 1/ε (fully polynomial...

Counting stable solutions of sparse polynomial systems in chemistry