Motivated by the concepts of the inverse Kazhdan--Lusztig polynomial and the equivariant Kazhdan--Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan--Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse ...

When do syzygies depend on the characteristic of the field? Even for well-studied families of examples, very little is known. For a family of random monomial ideals, namely, the Stanley--Reisner ideals of random flag complexes, we prove that the Betti ...

Given a parametric lattice with a basis given by polynomials in $\Bbb{Z}[t]$, we give an algorithm to construct an LLL-reduced basis whose elements are eventually quasi-polynomial in $t$: that is, they are given by formulas that are piecewise polynomial ...

We confirm a conjecture of Monical, Tokcan, and Yong on a characterization of the lattice points in the Newton polytopes of key polynomials.

We provide a symbolic trisection polynomial for Jacobians of genus 2 curves over finite field $\mathbb{F}_q$ of odd characteristic. These polynomials can be used to improve the efficiency of trisection algorithms, which may then be used to obtain faster ...

We prove that, for any positive $q$, the $q$-Eulerian polynomial of type $D$ has only real zeros. This settles an open problem of Brenti in 1994. For $q=1$, our result reduces to the real-rootedness of the Eulerian polynomials of type $D$, which was ...

In this paper, we investigate the permutation property of the Dickson polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$. Due to a known result, it suffices to consider permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$, where $p$ is a ...

We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal ...

We derive explicit formulas for the generating functions of the $q$-Bernstein basis functions in terms of $q$-exponential functions. Using these explicit formulas, we derive a collection of functional equations for these generating functions which we ...

For a given polynomial $P(X)$ of degree $d\ge 1$ modulo $p$, we estimate the number of elements $1\le u<p$ for which $P(u)$ coincides with the Fermat quotient $q_p(u)$ modulo $p$. In particular, the case $d=1$ improves the bound of Ostafe and Shparlinski on ...

Let $f : \mathbb{F}_q^l \rightarrow \mathbb{F}_q$ be a given function. We say that $f$ is a moderate expander if there exists an $\epsilon > 0$ such that for $|A| \gtrsim q^{1-\epsilon}$ one has that $|f(A,\ldots,A)| \gtrsim q$. We show that $x_1x_2 + (x_3-...

Phylogenetic tree reconstruction is a fundamental biological problem. Quartet amalgamation—combining a set of trees over four taxa into a tree over the full set—stands at the heart of many phylogenetic reconstruction methods. This task has attracted ...

Let $e_k(x)$ denote the $k$-th partial sum of the Maclaurin series for the exponential function. Define the $(n+1)\times(n+1)$ Hankel determinant by setting $\widetilde{H}_n(x)=\det[e_{i+j}(x)]_{0\leq i,j\leq n}$. We give a closed form evaluation of ...

This article uses combinatorial objects called labeled abaci to give direct combinatorial proofs of many familiar facts about Schur polynomials. We use abaci to prove the Pieri rules, the Littlewood-Richardson rule, the equivalence of the tableau ...

We present an efficient polynomial time approximation scheme (EPTAS) for scheduling on uniform processors, i.e., finding a minimum length schedule for a set of $n$ independent jobs on $m$ processors with different speeds (a fundamental NP-hard ...

In this paper, we consider Cusick's method to find the number of values of the polynomials $f_a(x)=x^{a}\,(x+1)^{2^k-1}$, when $x\in GF(2^{2k})$. We will prove that under certain conditions $f_a(x)$ and $f_{2-a}(x)$ have the same number of values. We ...

Based on a computer search, Anne Canteaut conjectured that the exponent $2^{2r}+2^r+1$ in ${\bf F}_{2^{6r}}$ and the exponent $(2^{r}+1)^2$ in ${\bf F}_{2^{4r}}$ yield bent monomial functions. These conjectures are proved in [A. Canteaut, P. Charpin, ...

Brooks proved that every connected graph other than a clique or odd cycle can be colored with $\Delta$ colors. Erdős, Rubin, and Taylor (and, independently, Vizing) generalized the theorem of Brooks to list colorings, describing all uncolorable ...

The concept of a jump system, introduced by Bouchet and Cunningham [SIAM J. Discrete Math., 8 (1995), pp. 17-32], is a set of integer points with a certain exchange property. In this paper, we discuss several linear and convex optimization problems on ...

The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded tree-width. The notion of clique-width extends the definition of cographs (graphs ...