Tropical multiplier sequences

TL;DR

This paper characterizes when diagonal operators defined by positive sequences preserve tropical and central indices, showing that log-concavity of the sequence is both necessary and sufficient, with implications for real-rooted polynomials.

Contribution

It provides a new characterization of log-concave sequences as the exact class preserving tropical and central indices under diagonal operators.

Findings

Diagonal operators preserve indices iff the sequence is log-concave.

Log-concavity is necessary and sufficient for preserving sign-independently real-rooted polynomials.

Elementary proof of the preservation of real-rootedness set by log-concave sequences.

Abstract

We prove that the diagonal operator defined by a positive sequence preserves tropical and central indices if and only if the sequence is log-concave. In particular we obtain an elementary proof of that such an operator preserves the set of sign-independently real-rooted polynomials if and only if the sequence is log-concave.