Induced log-concavity of equivariant matroid invariants

Inspired by the notion of equivariant log-concavity, we introduce the concept of induced log-concavity for a sequence of representations of a finite group. For an equivariant matroid equipped with a symmetric group action or a finite general linear group action, we transform the problem of proving the induced log-concavity of matroid invariants to that of proving the Schur positivity of symmetric functions. We prove the induced log-concavity of the equivariant Kazhdan-Lusztig polynomials of $q$-niform matroids equipped with the action of a finite general linear group, as well as that of the equivariant Kazhdan-Lusztig polynomials of uniform matroids equipped with the action of a symmetric group. As a consequence of the former, we obtain the log-concavity of Kazhdan-Lusztig polynomials of $q$-niform matroids, thus providing further positive evidence for Elias, Proudfoot and Wakefield's log-concavity conjecture on the matroid Kazhdan-Lusztig polynomials. From the latter we obtain the log-concavity of Kazhdan-Lusztig polynomials of uniform matroids, which was recently proved by Xie and Zhang by using a computer algebra approach. We also establish the induced log-concavity of the equivariant characteristic polynomials and the equivariant inverse Kazhdan-Lusztig polynomials for $q$-niform matroids and uniform matroids.Comment: 36 page

Factorization of non-negative operator valued trigonometric polynomials in two variables

Schur complements provide a convenient tool for proving the operator valued version of the classical (single variable) Fej\\u27er-Riesz problem. It also enables the factorization of multivariable trigonometric polynomials which are strictly positive. A result of Scheiderer implies that in two variables, nonnegative scalar valued trigonometric polynomials have sums of squares decompositions, Using a generalization of the notion of the Schur complement, we show how to extend this to operator valued trigonometric polynomials in two variables. We also indicate some other problems which may be tackled in a similar manner, including an operator version of Marshall\u27s Positivstellensatz on the strip

A bijective proof of Kohnert's rule for Schubert polynomials

Kohnert proposed the first monomial positive formula for Schubert polynomials as the generating polynomial for certain unit cell diagrams obtained from the Rothe diagram of a permutation. Billey, Jockusch and Stanley gave the first proven formula for Schubert polynomials as the generating polynomial for compatible sequences of reduced words of a permutation. In this paper, we give an explicit bijection between these two models, thereby definitively proving Kohnert's rule for Schubert polynomials.Comment: 8 pages, 7 figures (examples added, minor corrections

Chromatic Polynomials and Orbital Chromatic Polynomials and their Roots

The chromatic polynomial of a graph, is a polynomial that when evaluated at a positive integer k, is the number of proper k colorings of the graph. We can then find the orbital chromatic polynomial of a graph and a group of automorphisms of the graph, which is a polynomial whose value at a positive integer k is the number of orbits of k-colorings of a graph when acted upon by the group. By considering the roots of the orbital chromatic and chromatic polynomials, the similarities and differences of these polynomials is studied. Specifically we work toward proving a conjecture concerning the gap between the real roots of the chromatic polynomial and the real roots of the orbital chromatic polynomial

Formal Verification of Medina's Sequence of Polynomials for Approximating Arctangent

The verification of many algorithms for calculating transcendental functions is based on polynomial approximations to these functions, often Taylor series approximations. However, computing and verifying approximations to the arctangent function are very challenging problems, in large part because the Taylor series converges very slowly to arctangent-a 57th-degree polynomial is needed to get three decimal places for arctan(0.95). Medina proposed a series of polynomials that approximate arctangent with far faster convergence-a 7th-degree polynomial is all that is needed to get three decimal places for arctan(0.95). We present in this paper a proof in ACL2(r) of the correctness and convergence rate of this sequence of polynomials. The proof is particularly beautiful, in that it uses many results from real analysis. Some of these necessary results were proven in prior work, but some were proven as part of this effort.Comment: In Proceedings ACL2 2014, arXiv:1406.123