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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.

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