Real solutions to systems of polynomial equations in Macaulay2

TL;DR

The paper introduces the RealRoots package in Macaulay2 for analyzing real solutions of polynomial systems, including new methods, mathematical background, and a generalized Sylvester's Theorem.

Contribution

It updates and expands a previous package, providing new symbolic methods, mathematical insights, and a proof of a generalized Sylvester's Theorem.

Findings

Enhanced methods for real solution analysis

Mathematical background and illustrative examples

Proof of a generalized Sylvester's Theorem

Abstract

The Macaulay2 package RealRoots provides symbolic methods to study real solutions to systems of polynomial equations. It updates and expands an earlier package developed by Grayson and Sottile in 1999. We provide mathematical background and descriptions of the RealRoots package, giving examples which illustrate some of its implemented methods. We also prove a general version of Sylvester's Theorem whose statement and proof we could not find in the literature.