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

Abstract We define $i$-Lorentzian polynomials in two variables, and characterize them as the real homogeneous Macaulay dual generators whose corresponding codimension two algebras satisfy the mixed Hodge–Riemann relations in degree $i$ on the positive orthant of linear forms. We further show that $i$-Lorentzian polynomials of degree $d$ are in one-to-one correspondence with totally nonnegative Toeplitz matrices of size $(i+1)\times (d-i+1)$. Using this latter characterization, we show that a certain subclass of real rooted polynomials called normally stable are $i$-Lorentzian for all $i$. Our results also lead to a new theorem on Toeplitz matrices: the closure of the set of totally positive Toeplitz matrices equals the set of totally nonnegative Toeplitz matrices.