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We construct an effective algorithmic method to compute the homological monodromy of a complex polynomial which is tame. As an application we show the existence of conjugated polynomials in a number field which are not topologically…
We study a class of combinatorial scheduling problems characterized by a particular type of constraint often associated with electrical power or gas energy. This constraint appears in several practical applications and is expressed as a sum…
An interesting observation is that most pairs of weakly homogeneous mappings have no strongly monotonic property, which is one of the key conditions to ensure the unique solvability of the generalized variational inequality. This paper…
In this paper we illustrate that paraorthogonality on the unit circle $\mathbb{T}$ is the counterpart to orthogonality on $\mathbb{R}$ when we are interested in the spectral properties. We characterize quasi-paraorthogonal polynomials on…
In this article we describe a new inductive approach to compute the chromatic polynomial of simple graphs and the characteristic polynomial of central hyperplane arrangements.
We introduce the combinatorial Lyubeznik resolution of monomial ideals. We prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As an application, we give a combinatorial method to determine if an ideal is a…
We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with…
We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The…
We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…
We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive…
We consider the problem of extending the classical S-lemma from commutative case to noncommutative cases. We show that a symmetric quadratic homogeneous matrix-valued polynomial is positive semidefinite if and only if its coefficient matrix…
We study the theory of equations in one variable over polyhedral semirings. The article revolves around a notion of solution to a polynomial equation over a polyhedral semiring. Our main results are a characterisation of local solutions in…
In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k…
This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like…
We consider the distinct elements problem, where the goal is to estimate the number of distinct colors in an urn containing $ k $ balls based on $n$ samples drawn with replacements. Based on discrete polynomial approximation and…
We study the notion of regular singularities for parameterized complex ordinary linear differential systems, prove an analogue of the Schlesinger theorem for systems with regular singularities and solve both a parameterized version of the…
In recent years, the notion of characteristic polynomial of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the…
We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a H\"older continuous linear term. With the help of those formulas we are able to…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
We introduce a new concept of infinite quasi-exactly solvable models which are constructable through multi-parameter deformations of known exactly solvable ones. The spectral problem for these models admits exact solutions for infinitely…