Related papers: Integer decomposition property of dilated polytope…

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…

We compute the deformation space of quadratic letterplace ideals $L(2,P)$ of finite posets $P$ when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The…

Recently, Hajdu and S\'{a}rk\"{o}zy studied the multiplicative decompositions of polynomial sequences. In particular, they showed that when $k \geq 3$, each infinite subset of $\{x^k+1: x \in \mathbb{N}\}$ is multiplicatively irreducible.…

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive…

We describe an efficient algorithm to compute finite type invariants of type $k$ by first creating, for a given knot $K$ with $n$ crossings, a look-up table for all subdiagrams of $K$ of size $\lceil \frac{k}{2}\rceil$ indexed by dyadic…

A lattice polytope $P$ is called IDP if any lattice point in its $k$th dilate is a sum of $k$ lattice points in $P$. In 1991 Stanley proved a strong inequality in Ehrhart theory for IDP lattice polytopes. We show that his conclusion holds…

In [2], I constructed the p-adic q-integral on Zp. In this paper, we consider the properties of the p-adic invariant p-adic q-integral in the ring of p-adic integers at q=-1. Finally we give the some applications of p-adic q-integration at…

In this note we consider the question how the set of inversions of a permutation $\pi \in S_n$ can be partitioned into two subset, such that those are itself inversion sets of permutations. This is archived by exploiting a connection to a…

The computation of Kronecker coefficients is a challenging problem with a variety of applications. In this paper we present an approach based on methods from symplectic geometry and residue calculus. We outline a general algorithm for the…

Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <=…

In this paper we deal with edge-to-edge, irreducible decompositions of a centrally symmetric convex $(2k)$-gon into centrally symmetric convex pieces. We prove an upper bound on the number of these decompositions for any value of $k$, and…

We introduce an extension, indexed by a partially ordered set P and cardinal numbers k,l, denoted by (k,l)-->P, of the classical relation (k,n,l)--> r in infinite combinatorics. By definition, (k,n,l)--> r holds, if every map from the…

We prove that octants are cover-decomposable into multiple coverings, i.e., for any k there is an m(k) such that any m(k)-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into k…

Any complex-valued polynomial on $(\mathbb{R}^n)^k$ decomposes into an algebraic combination of $O(n)$-invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if $n \geq 2k-1$. We…

This paper is concerned with the polynomial integrability of the two-dimensional Hamiltonian systems associated to complex homogeneous polynomial potentials of degree $k$ of type $V_{k,l}=\alpha (q_2-i q_1)^l (q_2+iq_1)^{k-l}$ with…

The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in…

The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \begin{align*} \sum_{k=0}^{p-1}(2k+1)D_k(x)^3…

Let $l\geq 6$ be any integer, where $l\equiv 2$ mod $4$. Suppose that $\mu(\tau)d\tau$ is a measure with bounded variation and is supported on a compact subset of the complex plane, where…

We present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal…

Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a…