Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann Property for Graded Oriented Artinian Gorenstein Algebras in Codimension Two

Abstract

A (standard graded) oriented Artinian Gorenstein algebra over the real numbers is uniquely determined by a real homogeneous polynomial called its Macaulay dual generator. We study the mixed Hodge-Riemann relations on oriented Artinian Gorenstein algebras for which we give a signature criterion on the higher mixed Hessian matrices of its Macaulay dual generator. Inspired by recent work of Br\"and\'en and Huh, we introduce a class of homogeneous polynomials in two variables called $i$-Lorentzian polynomials, and show that these are exactly the Macaulay dual generators of oriented Artinian Gorenstein algebras in codimension two satisfying mixed Hodge-Riemann relations up to degree $i$ on the positive orthant of linear forms. We further show that the set of $i$-Lorentzian polynomials of degree $d$ are in one-to-one correspondence with the set of totally nonnegative Toeplitz matrices of size depending on $i$ and $d$. A corollary is that all normally stable polynomials, i.e. polynomials whose normalized coefficients form a PF sequence, are $i$-Lorentzian. Another corollary is an analogue of Whitney's theorem for Toeplitz matrices, which appears to be new: the closure of the set of totally positive Toeplitz matrices, in the Euclidean space of all real matrices of a given size, is equal to the set of totally nonnegative Toeplitz matrices.