Related papers: An excursion from enumerative goemetry to solving …
A survey written for the upcoming "Handbook of Enumerative Combinatorics".
In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii) Geometry of Polynomials Level Sets, and iii) Shape…
We discuss the use of methods coming from integrable systems to study problems of enumerative and algebraic combinatorics, and develop two examples: the enumeration of Alternating Sign Matrices and related combinatorial objects, and the…
We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and…
Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…
Enumerative kernelization is a recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms. Its study began with the paper of Creignou et al. [Theory Comput. Syst., 2017], and development in…
Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent…
In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic surfaces. The series expansions originate at the intersection of the surface with as many coordinate planes as the dimension of the surface.…
Although the P\'olya enumeration theorem has been used extensively for decades, an optimized, purely numerical algorithm for calculating its coefficients is not readily available. We present such an algorithm for finding the number of…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…
Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and…
Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower…
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…
In recent developments, a general approach for solving Riemann--Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this…
We elaborate on the connection between Gel'fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman integrals. We propose a novel, more efficient algorithm to compute Macaulay…
We discuss the problem of whether a given problem in enumerative geometry can have all of its solutions be real. In particular, we describe an approach to problems of this type, and show how this can be used to show some enumerative…
This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on…
Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry…
Symmetry in mathematical programming may lead to a multiplicity of solutions. In nonconvex optimisation, it can negatively affect the performance of the branch-and-bound algorithm. Symmetry may induce large search trees with multiple…