Corrigendum to "Higher Lorentzian polynomials,...in codimension two" [International Mathematics Research Notices, Volume 2025, Issue 13, July 2025, arXiv:2208.05653]

TL;DR

This paper corrects a previous result by proving that total nonnegativity of Toeplitz matrices associated with bivariate forms is preserved when reducing the matrix size, using Hodge theory and Schur polynomials.

Contribution

It provides a corrected proof establishing the preservation of total nonnegativity in Toeplitz matrices derived from bivariate forms, filling a gap in earlier work.

Findings

Total nonnegativity is preserved when reducing Toeplitz matrix size.

Hodge theory and Schur polynomials are effective tools for this proof.

The correction solidifies the theoretical foundation of the original result.

Abstract

A homogeneous bivariate $d$-form defines an $(i+1)$-rowed Toeplitz matrix for each $i$ between $0$ and $d$. We use Hodge theory and Schur polynomials to prove that if the $(i+1)$-rowed Toeplitz matrix of a form is totally nonnegative, then so is the $i$-rowed one. This fixes a gap in the main result of paper above.