Related papers: Using monodromy to recover symmetries of polynomia…
The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the…
We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic…
We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the…
We develop a certified numerical algorithm for computing Galois/monodromy groups of parametrized polynomial systems. Our approach employs certified homotopy path tracking to guarantee the correctness of the monodromy action produced by the…
We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows a framework developed in previous work and can operate in…
The Galois group of a parameterized polynomial system of equations encodes the structure of the solutions. This monodromy group acts on the set of solutions for a general set of parameters, that is, on the fiber of a projection from the…
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…
Synthesis problems for linkages in kinematics often yield large structured parameterized polynomial systems which generically have far fewer solutions than traditional upper bounds would suggest. This paper describes statistical models for…
We introduce the concept of monodromy coordinates for representing solutions to large polynomial systems. Representing solutions this way provides a time-memory trade-off in a monodromy solving algorithm. We describe an algorithm, which…
By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure…
We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because…
We propose solving the power flow equations using monodromy. We prove the variety under consideration decomposes into trivial and nontrivial subvarieties and that the nontrivial subvariety is irreducible. We also show various symmetries in…
We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…
Motivated by applications of algebraic geometry, we introduce the Galois width, a quantity characterizing the complexity of solving algebraic equations in a restricted model of computation allowing only field arithmetic and adjoining…
We explore the enumerative problem of finding lines on cubic surfaces defined by symmetric polynomials. We prove that the moduli space of symmetric cubic surfaces is an arithmetic quotient of the complex hyperbolic line, and determine…
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…
Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and…
We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear…
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…
In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in…