# On Descartes' rule of signs

## TL;DR

This paper investigates the realizability of specific sign patterns and root sign distributions in real polynomials, providing new conditions for cases with exactly two sign changes, extending Descartes' rule of signs.

## Contribution

It offers new sufficient conditions for the realizability of root sign pairs in polynomials with exactly two sign changes, advancing understanding of Descartes' rule of signs.

## Key findings

- Identifies conditions for the realizability of certain root sign pairs
- Extends Descartes' rule of signs to specific cases with two sign changes
- Provides criteria for polynomial sign pattern realizability

## Abstract

A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum _{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP $\sigma :=(+$,sgn$(a_{d-1})$, $\ldots$, sgn$(a_0))$. By Descartes' rule of signs, for the quantity $pos$ of positive (resp. $neg$ of negative) roots of $P$, one has $pos\leq c$ (resp. $neg\leq p=d-c$), where $c$ and $p$ are the numbers of sign changes and sign preservations in $\sigma$; the numbers $c-pos$ and $p-neg$ are even. We say that $P$ realizes the SP $\sigma$ with the pair $(pos, neg)$. For SPs with $c=2$, we give some sufficient conditions for the (non)realizability of pairs $(pos, neg)$ of the form $(0,d-2k)$, $k=1$, $\ldots$, $[(d-2)/2]$.