Proof of a conjecture on unimodality

TL;DR

This paper proves a conjecture that shifting a polynomial with nonnegative, non-decreasing coefficients by any positive real number results in a unimodal sequence of coefficients, extending previous integer-shift results.

Contribution

It establishes the unimodality of polynomial coefficients after real shifts and analyzes the modes' locations, providing new conditions for unimodality and mode uniqueness.

Findings

Coefficients of P(x+d) are unimodal for any positive real d.

The paper identifies conditions for P(x+d) to have a unique mode.

It generalizes previous results from integer to real shifts.

Abstract

Let $P(x)$ be a polynomial of degree $m$, with nonnegative and non-decreasing coefficients. We settle the conjecture that for any positive real number $d$, the coefficients of $P(x+d)$ form a unimodal sequence, of which the special case $d$ being a positive integer has already been asserted in a previous work. Further, we explore the location of modes of $P(x+d)$ and present some sufficient conditions on $m$ and $d$ for which $P(x+d)$ has the unique mode $\lceil{m-d\over d+1}\rceil$.