Evaluating geometric queries using few arithmetic operations

TL;DR

This paper presents a method to evaluate the signs of polynomial families efficiently using few arithmetic operations, with a database structure enabling quick sign queries for points in a bounded domain.

Contribution

It introduces a novel algebraic computation tree-based database for sign evaluation of polynomial systems, optimizing the number of arithmetic operations needed.

Findings

Efficient sign determination with $h d^{O(n^2)}$ comparisons.

Construction of a database supporting rapid sign queries.

Application to polynomial system consistency checking with minimal arithmetic operations.

Abstract

Let $\cp:=(P_1,...,P_s)$ be a given family of $n$-variate polynomials with integer coefficients and suppose that the degrees and logarithmic heights of these polynomials are bounded by $d$ and $h$, respectively. Suppose furthermore that for each $1\leq i\leq s$ the polynomial $P_i$ can be evaluated using $L$ arithmetic operations (additions, subtractions, multiplications and the constants 0 and 1). Assume that the family $\cp$ is in a suitable sense \emph{generic}. We construct a database $\cal D$, supported by an algebraic computation tree, such that for each $x\in [0,1]^n$ the query for the signs of $P_1(x),...,P_s(x)$ can be answered using $h d^{\cO(n^2)}$ comparisons and $nL$ arithmetic operations between real numbers. The arithmetic-geometric tools developed for the construction of $\cal D$ are then employed to exhibit example classes of systems of $n$ polynomial equations in $n$ unknowns whose consistency may be checked using only few arithmetic operations, admitting however an exponential number of comparisons.