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We consider the problem of computing the Euler characteristic of an abstract simplicial complex given by its vertices and facets. We show that this problem is #P-complete and present two new practical algorithms for computing Euler…

In the present paper, we provide a cohomology group as a categorification of the characteristic polynomial of matroids. The construction depends on the ``quasi-representation'' of a matroid. For a certain choice of the quasi-representation,…

We present a number of combinatorial characterizations of K-matrices. This extends a theorem of Fiedler and Ptak on linear-algebraic characterizations of K-matrices to the setting of oriented matroids. Our proof is elementary and simplifies…

Binary polynomial optimization is equivalent to the problem of minimizing a linear function over the intersection of the multilinear set with a polyhedron. Many families of valid inequalities for the multilinear set are available in the…

The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The…

Given an irreducible hypersurface singularity of dimension $d$ (defined by a polynomial $f\in K[[ {\bf x} ]][z]$) and the projection to the affine space defined by $K[[ {\bf x} ]]$, we construct an invariant which detects whether the…

Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic…

In this paper we introduce a class of polygonal complexes for which we can define a notion of sectional combinatorial curvature. These complexes can be viewed as generalizations of 2-dimensional Euclidean and hyperbolic buildings. We focus…

A. Gasull shared a list of 33 open problems in low dimensional dynamical systems in his work in 2021. The second part of Problem 3 is about whether the limit cycle of a quasi-homogeneous system $ \dot{x}=y,\; \dot{y}=-x^3+\alpha x^2y+y^3 $…

In arXiv:0810.2076 we presented a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the representation varieties of Riemann surfaces with semi-simple conjugacy…

There are two well known tasks, related to Newton polyhedra: to study invariants of singularities in terms of their Newton polyhedra, and to describe Newton polyhedra of resultants and discriminants. We introduce so called resultantal…

We conjecture a new ordinary differential equation exactly isospectral to the radial component of the homogeneous Teukolsky equation. We find this novel relation by a hidden symmetry implied from a four-dimensional $\mathcal{N}=2$…

This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with…

We survey three methods for proving that the characteristic polynomial of a finite lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on…

We give a formula for computing the characteristic polynomial for certain hyperplane arrangements in terms of the number of bipartite graphs of given rank and cardinality.

In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of $E_7$ into a sum of irreducible submodules. Moreover, we obtain a combinatorial identity, saying…

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…

In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with…

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The…

Characteristic quasi-polynomials are the enumerative functions counting the number of elements in the complement of hyperplane arrangements modulo positive integers. A notable phenomenon in this context is period collapse, where the…