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Creators/Authors contains: "Macias-Marques, Pedro"

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  1. ; ; (, International Mathematics Research Notices)
    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. 
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    Free, publicly-accessible full text available July 1, 2026
  2. ; ; ; ; (, International Mathematics Research Notices)
    Abstract We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript. 
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