On the Modes of Polynomials Derived from Nondecreasing Sequences

TL;DR

This paper proves a conjecture regarding the behavior of the smallest and greatest modes of polynomials with nondecreasing nonnegative coefficients when shifted by different positive values, confirming that these modes decrease as the shift increases.

Contribution

The paper provides a proof for Wang and Yeh's conjecture on the monotonicity of polynomial modes under shifts for polynomials with nondecreasing nonnegative coefficients.

Findings

Confirmed that $M_*(P,d)$ decreases as $d$ increases.

Confirmed that $M^*(P,d)$ decreases as $d$ increases.

Validated the conjecture for all such polynomials.

Abstract

Wang and Yeh proved that if $P(x)$ is a polynomial with nonnegative and nondecreasing coefficients, then $P(x+d)$ is unimodal for any $d>0$. A mode of a unimodal polynomial $f(x)=a_0+a_1x+\cdots + a_mx^m$ is an index $k$ such that $a_k$ is the maximum coefficient. Suppose that $M_*(P,d)$ is the smallest mode of $P(x+d)$, and $M^*(P,d)$ the greatest mode. Wang and Yeh conjectured that if $d_2>d_1>0$, then $M_*(P,d_1)\geq M_*(P,d_2)$ and $M^*(P,d_1)\geq M^*(P,d_2)$. We give a proof of this conjecture.