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1. Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

   https://doi.org/10.1142/S0219498821400168  

   Riley Casper, W.; Kolb, Stefan; Yakimov, Milen (January 2021, Journal of Algebra and Its Applications)

   null (Ed.)

   To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials. 

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

   https://doi.org/10.1093/imrn/rnaf183  

   Macias-Marques, Pedro; McDaniel, Chris; Seceleanu, Alexandra (July 2025, 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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3. Low Rank Tensor Decompositions and Approximations

   https://doi.org/10.1007/s40305-023-00455-7  

   Nie, Jiawang; Wang, Li; Zheng, Zequn (March 2023, Journal of the Operations Research Society of China)

   Abstract There exist linear relations among tensor entries of low rank tensors. These linear relations can be expressed by multi-linear polynomials, which are called generating polynomials. We use generating polynomials to compute tensor rank decompositions and low rank tensor approximations. We prove that this gives a quasi-optimal low rank tensor approximation if the given tensor is sufficiently close to a low rank one. 

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4. Motives of melonic graphs

   https://doi.org/10.4171/AIHPD/156  

   Aluffi, Paolo; Marcolli, Matilde; Qaisar, Waleed (July 2023, Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions)

   We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-4internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space\mathcal M_{0,4}. We also conjecture that the corresponding polynomials arelog-concave, on the basis of hundreds of explicit computations. 

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5. Hodge–Riemann Relations for Schur Classes in the Linear and Kähler Cases

   https://doi.org/10.1093/imrn/rnac208  

   Ross, Julius; Toma, Matei (August 2022, International Mathematics Research Notices)

   We prove a version of the Hodge–Riemann bilinear relations for Schur polynomials of Kähler forms and for Schur polynomials of positive forms on a complex vector space. 

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