2022/08/11 by Marques, Pedro Macias, McDaniel, Chris, Seceleanu, Alexandra +1
#11B83 #13E10 #13H10 #14F45 #15B05 #15B35 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.05653
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 Brändén 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.