Descartes's "Rule of Signs'' and Poincar\'e's Positivstellensatz

TL;DR

This paper explores Poincaré's refinement of Descartes's rule of signs, highlighting its significance as a precursor to Pólya's Positivstellensatz and its impact on algebraic equations analysis.

Contribution

It provides an exposition of Poincaré's 1883 work, clarifying its role as a refinement of Descartes's rule and its influence on the development of Positivstellensatz.

Findings

Refinement of Descartes's rule of signs by Poincaré

Connection between Poincaré's work and Pólya's Positivstellensatz

Historical insight into algebraic equations analysis

Abstract

This is an exposition of Poincar\'e's 1883 paper, ``Sur les \'equations alg\'ebriques,'' which gives an important refinement of Descartes's rule of signs and was a precursor of P\' olya's Positivstellensatz.