Related papers: Certifying Galois/monodromy Actions via Homotopy G…

A path tracking algorithm that adaptively adjusts precision is presented. By adjusting the level of precision in accordance with the numerical conditioning of the path, the algorithm achieves high reliability with less computational cost…

We develop a Galois theory for systems of linear difference equations with an action of an endomorphism {\sigma}. This provides a technique to test whether solutions of such systems satisfy {\sigma}-polynomial equations and, if yes, then…

We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular…

In order to apply canonical labelling of graphs and isomorphism checking in interactive theorem provers, these checking algorithms must either be mechanically verified or their results must be verifiable by independent checkers. We analyze…

We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by S. Mori and study the monodromy of corresponding hyperelliptic jacobians.

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…

In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant's model. We are given a fixed graph $H$ and want to find all graphs, from some graph class,…

We present a new algorithm to compute all the chiral polytopes that have a given group $G$ as full automorphism group. This algorithm uses a new set of generators that characterize the group, all of them except one being involutions. It…

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate…

We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of their Galois groups as subgroups of…

Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous…

The monodromy group is an invariant for parameterized systems of polynomial equations that encodes structure of the solutions over the parameter space. Since the structure of real solutions over real parameter spaces are of interest in many…

We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients…

In the recent years, several polynomial algorithms of a dynamical nature have been proposed to address the graph isomorphism problem. In this paper we propose a generalization of an approach exposed in cond-mat/0209112 and find that this…

Combining ideas from distributed algorithms and alternating automata, we introduce a new class of finite graph automata that recognize precisely the languages of finite graphs definable in monadic second-order logic. By restricting…

We use our recent implementation of a certified homotopy tracking algorithm to search for start systems that minimize the average complexity of finding all roots of a regular system of polynomial equations. While finding optimal start…

We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible…

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…

The goal of this paper is to study the path-following method for univariate polynomials. We propose to study the complexity and condition properties when the Newton method is applied as a correction operator. Then we study the geodesics and…

We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of…