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1. Some Remarks on Non-Symmetric Interpolation Macdonald Polynomials

   https://doi.org/10.1093/imrn/rnz229  

   Sahi, Siddhartha; Stokman, Jasper (November 2019, International Mathematics Research Notices)

   Abstract We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular to binomial formulas involving non-symmetric interpolation Macdonald polynomials. 

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2. Combinatorial anti-concentration inequalities, with applications

   https://doi.org/10.1017/S0305004120000183  

   FOX, JACOB; KWAN, MATTHEW; SAUERMANN, LISA (September 2021, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/ e + o (1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs. 

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3. Some Properties and Invariants of Multivariate Difference-Differential Dimension Polynomials.

   Levin, A. (June 2018, Proceedings of the 24th Conference on Applications of Computer Algebra - ACA 2018)

   Multivariate dimension polynomials associated with finitely generated differential and difference field extensions arise as natural generalizations of the univariate differential and difference dimension polynomials. It turns out, however, that they carry more information about the corresponding extensions than their univariate counterparts. We extend the known results on multivariate dimension polynomials to the case of difference-differential field extensions with arbitrary partitions of sets of basic operators. We also describe some properties of multivariate dimension polynomials and their invariants. 

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4. Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

   https://doi.org/10.1093/imrn/rnac291  

   Blekherman, Grigoriy; Kummer, Mario; Sanyal, Raman; Shu, Kevin; Sun, Shengding (November 2022, International Mathematics Research Notices)

   Abstract A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection with multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to find an explicit degree 3 hyperbolic polynomial in six variables some of whose Rayleigh differences are not sums of squares. We further generalize the well-known Fisher–Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment. 

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5. Dimension Polynomials and the Einstein’s Strength of Some Systems of Quasi-linear Algebraic Difference Equations

   Evgrafov, A.; Levin, A. (June 2018, Proceedings of the 24th Conference on Applications of Computer Algebra - ACA 2018)

   We present a difference algebraic technique for the evaluation of the Einstein's strength of quasi-linear partial difference equations and some systems of such equations. Our approach is based on the properties of difference dimension polynomials that express the Einstein's strength and on the characteristic set method for computing such polynomials. The obtained results are applied to the comparative analysis of difference schemes for some chemical reaction-diffusion equations. 

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