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

Pedro Macias Marques, Chris McDaniel, Alexandra Seceleanu

TL;DR

This corrigendum fixes a gap in MMS Theorem 2 by establishing a precise network of equivalences among $i$-Lorentzian forms, strong total nonnegativity of Toeplitz matrices $oldsymbol{\phi}^i_d(F)$, and the mixed Hodge–Riemann relations on the standard open cone $U$ via Schur polynomials and the Littlewood–Richardson rule. It develops a Plücker-type expansion for mixed Hessians and a Schur-polynomial expansion for Toeplitz minors, then uses weighted NE lattice paths and LGV to prove positivity and a downward-induction argument to transfer Lorentzian properties to HRR. Consequently, it shows that total nonnegativity of $oldsymbol{\phi}^i_d(F)$ implies mixed HRR$_i$, and that strong total nonnegativity and total nonnegativity coincide (STN=TN) for these Toeplitz matrices, thereby completing the MMS program and clarifying the normally stable MMS subfamily. The corrigendum also provides detailed errata corrections to MMS and clarifies several structural aspects linking Lorentzian polynomials, Toeplitz matrices, and Schur-positivity within the Hodge–Riemann framework.

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.

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

TL;DR

This corrigendum fixes a gap in MMS Theorem 2 by establishing a precise network of equivalences among -Lorentzian forms, strong total nonnegativity of Toeplitz matrices , and the mixed Hodge–Riemann relations on the standard open cone via Schur polynomials and the Littlewood–Richardson rule. It develops a Plücker-type expansion for mixed Hessians and a Schur-polynomial expansion for Toeplitz minors, then uses weighted NE lattice paths and LGV to prove positivity and a downward-induction argument to transfer Lorentzian properties to HRR. Consequently, it shows that total nonnegativity of implies mixed HRR, and that strong total nonnegativity and total nonnegativity coincide (STN=TN) for these Toeplitz matrices, thereby completing the MMS program and clarifying the normally stable MMS subfamily. The corrigendum also provides detailed errata corrections to MMS and clarifies several structural aspects linking Lorentzian polynomials, Toeplitz matrices, and Schur-positivity within the Hodge–Riemann framework.

Abstract

A homogeneous bivariate -form defines an -rowed Toeplitz matrix for each between and . We use Hodge theory and Schur polynomials to prove that if the -rowed Toeplitz matrix of a form is totally nonnegative, then so is the -rowed one. This fixes a gap in the main result of paper above.
Paper Structure (9 sections, 11 theorems, 66 equations, 3 figures)

This paper contains 9 sections, 11 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Toeplitz Fundamentals
  3. Schur polynomials and totally nonnegative Toeplitz matrices
  4. A Primer on Schur Polynomials
  5. From Toeplitz minors to Schur polynomials
  6. Mixed Hessian determinants and weighted NE lattice paths
  7. Proof of Theorem \ref{['thm:Lorentzian2']}
  8. Proof of Theorem \ref{['thm:STN=TN']}
  9. Other Errata

Key Result

Theorem 1.1

Let $F\in Q_d$ be a homogeneous $d$-form, and fix $0\leq i\leq \left\lfloor\frac{d}{2}\right\rfloor$. The following are equivalent.

Figures (3)

  • Figure 1: Translating a consecutive submatrix of $\phi^{i-1}_d(F)$ into an initial submatrix of $\phi^i_d(F)$
  • Figure 2: The unique $\nu$-filling of the skew shape $\lambda/(\mu+a)$. The dots in the lower right corner count the $\alpha^i$ statistic.
  • Figure 3: vertex disjoint path system for the monomial term $X_1^2X_2^2Y_1Y_2$ of $\Delta_{(K+r)J}(W_2(X_1,X_2,Y_1,Y_2))$ for $K+r=\{3,4,6\}$ and $J=\{2,3,4\}$

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.4
  • ...and 15 more