Related papers: Three Combinatorial Algorithms for the Cave Polyno…
Many mathematicians have been studying various degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our main focus here is a new type of degenerate poly-Euler polynomials and numbers. This…
A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given. The main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.
In this paper we present recent results on parametric analysis of biological models. The underlying method is based on the algorithms for computing trajectory sets of hybrid systems with polynomial dynamics. The method is then applied to…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
We give an effective method to compute the entropy for polynomials orthogonal on a segment of the real axis that uses as input data only the coefficients of the recurrence relation satisfied by these polynomials. This algorithm is based on…
Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among…
In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that…
Consider an algebraic equation $P(x,y)=0$ where $P\in \mathbb C[x,y] $ (or $\mathbb F[x,y]$ with $\mathbb F\subset \mathbb C$ a subfield) is a bivariate polynomial, it defines a plane algebraic curve. We provide an efficient method for…
Multimatroids generalize matroids, delta-matroids, and isotropic systems, and transition polynomials of multimatroids subsume various polynomials for these latter combinatorial structures, such as the interlace polynomial and the…
OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number…
The purpose of this work is to introduce a strategy for determining the nodes and weights of a low-cardinality positive cubature formula nearly exact for polynomials of a given degree over spherical polygons. In the numerical section we…
Previous work has made use of a parameterized plane curve polynomial representation for mathematical handwriting, with the polynomials represented in a Legendre or Legendre-Sobolev graded basis. This provides a compact geometric…
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…
For Paley-Wiener functions on weighted combinatorial finite or infinite graphs we develop a weighted sampling theory in which samples are defined as inner products with weight functions (measuring devices). Three reconstruction methods are…
Let us consider a polynomial algebra in three variables equipped with an integer grading. We construct a system of group-generating automorphisms that preserve a given grading.
We provide a natural definition of an elliptic arrangement, extending the classical framework to an elliptic curve E with complex multiplication. We analyse the intersections of elements of the arrangement and their connected components as…
A family of polynomials parameterized by the conjugacy classes of a finite Coxeter group is investigated. These polynomials, together with the character table of the group, determine the associated generic degrees. The polynomials are…
In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the…
We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…
Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and…