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

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.

💡 Research Summary

The corrigendum addresses a subtle but crucial gap in the proof of the main result of the 2025 IMRN paper “Higher Lorentzian polynomials, higher Hessians, and the Hodge‑Riemann relations for graded oriented Artinian Gorenstein algebras in codimension two”. The original argument claimed that if the (i + 1)-row Toeplitz matrix associated to a homogeneous bivariate d‑form F is totally non‑negative (TNN), then the i‑row Toeplitz matrix is also TNN, and from this deduced the equivalence between “strongly TNN” and “TNN”. However, the proof relied on the very equivalence it was trying to establish, creating a circular reasoning problem.

The authors resolve this by developing two complementary tools. First, they study the combinatorial structure of Toeplitz matrices. Lemma 2.1 shows that any consecutive minor of the i‑row Toeplitz matrix ϕ_i^d(F) can be shifted to an initial minor, and that the set of consecutive minors of ϕ_{i‑1}^d(F) coincides with the set of consecutive minors of ϕ_i^d(F) of size ≤ i. This allows one to reduce questions about total positivity (TP) or total non‑negativity to the behavior of consecutive minors only.

Second, they bring in the theory of Schur polynomials and the Littlewood‑Richardson rule. Using the generalized Jacobi‑Trudi identity (Fact 3.2), any (r + 1) × (r + 1) minor of a Toeplitz matrix can be expressed as a determinant of complete symmetric functions, which in turn equals a Schur polynomial s_ν evaluated at a suitable point (Equation (3)–(4)). The partition ν is determined by the column index set of the minor. Because Schur polynomials form a basis of symmetric functions, the coefficients in their expansion are precisely the Littlewood‑Richardson numbers c_{λµ}^ν, which are known to be non‑negative integers.

The key observation is that if a Toeplitz minor is non‑negative, then the corresponding Schur polynomial evaluates to a non‑negative number. By the Littlewood‑Richardson rule this forces all relevant Littlewood‑Richardson coefficients to be non‑negative, which in turn guarantees the non‑negativity of every consecutive minor of the Toeplitz matrix. Consequently, the matrix is TNN. This argument works without assuming any prior equivalence between strong and ordinary total non‑negativity.

Lemma 2.2 establishes that for totally positive (TP) Toeplitz matrices, strong total positivity (all consecutive minors positive) is equivalent to ordinary total positivity, using Fekete’s theorem that positivity of all consecutive minors implies total positivity. Lemma 2.4 extends the analysis to the non‑negative case by introducing the Sperner number s(F) (the maximal dimension of the graded Artinian Gorenstein algebra A_F). It shows that if the (s − 1)-row Toeplitz matrix is TP, then every Toeplitz matrix ϕ_j^d(F) with j ≥ s is TP of order s and simultaneously TNN, thereby linking strong TNN and ordinary TNN.

With these technical foundations, the authors prove two main theorems. Theorem 1.1 states the equivalence of three conditions for a homogeneous form F and a fixed i ≤ ⌊d/2⌋: (1) F is i‑Lorentzian; (2) the i‑row Toeplitz matrix ϕ_i^d(F) is strongly totally non‑negative; (3) the Artinian Gorenstein algebra A_F satisfies the mixed Hodge‑Riemann relations HRR_i on the standard open cone U = {ax + by | a,b > 0}. Theorem 1.2 shows that if ϕ_i^d(F) is merely TNN (not necessarily strong), then A_F still satisfies the mixed HRR_i on U. Together these results fill the logical gap in the original paper: they prove that strong total non‑negativity and total non‑negativity are indeed equivalent for Toeplitz matrices arising from bivariate forms, and thus validate the original Theorem 4.21.

The final section lists minor typographical corrections and clarifies notation, ensuring consistency with the original work. Overall, the corrigendum provides a rigorous, self‑contained proof that the hierarchy of Toeplitz matrices respects total non‑negativity, bridges combinatorial Schur‑function theory with Hodge‑Riemann geometry, and restores the validity of the main results of the 2025 IMRN article.