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In this paper, we study the desingularization problem in the first q -Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first q -Weyl algebra. Moreover, an algorithm is presented for computing a generating set of the first q -Weyl closure of a given q -difference operator. As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator. [ABSTRACT FROM AUTHOR]

Abstract: In an earlier paper with Whittle, we showed that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid M. The purpose of this paper is to give a polynomial-time algorithm for constructing such a tree for M. [Copyright &y& Elsevier]

Abstract: In a recent paper (Cucker et al., 2008 ) we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number for the input system f. In this paper we look at as a random variable derived from imposing a probability measure on the space of polynomial systems and give bounds for both the tail and the expected value . [Copyright &y& Elsevier]

Abstract: This paper presents a new algorithm for the absolute factorization of parametric multivariate polynomials over the field of rational numbers. This algorithm decomposes the parameters space into a finite number of constructible sets. The absolutely irreducible factors of the input parametric polynomial are given uniformly in each constructible set. The algorithm is based on a parametric version of Hensel''s lemma and an algorithm for quantifier elimination in the theory of algebraically closed field in order to reduce the problem of finding absolute irreducible factors to that of representing solutions of zero-dimensional parametric polynomial systems. The complexity of this algorithm is single exponential in the number n of the variables of the input polynomial, its degree d w.r.t. these variables and the number r of the parameters. [Copyright &y& Elsevier]

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4]. Let U m , d denote the uniform matroid of rank d on a set of m + d elements. Gedeon et al. (2017) [7] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of U m , d using equivariant Kazhdan-Lusztig polynomials. In this paper we give an alternative explicit formula, which allows us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of U m , d for 2 ≤ m ≤ 15 and all d 's. The case m = 1 was previously proved by Gedeon et al. (2017) [8]. We further determine the Z -polynomials of all U m , d 's and prove the real-rootedness of the Z -polynomials of U m , d for 2 ≤ m ≤ 15 and all d 's. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Young's formula for the Kazhdan-Lusztig polynomials of U m , d 's without using the equivariant Kazhdan-Lusztig polynomials. [ABSTRACT FROM AUTHOR]