The problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain. Proving Rouché’s Theorem by homotopy continuation concepts yields a new class of homotopy methods, the socalled regional homotopy methods. These methods rely on isolating a part of the system to be solved, which dominates the rest of the system on the border of the domain. ...

Polyhedral homotopy continuation methods exploit the sparsity of polynomial systems so that the number of solution curves to reach all isolated solutions is optimal for generic systems. The numerical stability of tracing solution curves of polyhedral homotopies is mainly determined by the height of the powers of the continuation parameter. To reduce this height we propose a procedure that opera...

Numerical algebraic geometry is the area devoted to the solution and manipulation of polynomial systems by numerical methods, which are mainly based on continuation. Due to the extreme intrinsic parallelism of continuation, polynomial systems may be successfully dealt with that are much larger than is possible with other methods. Singular solutions require special numerical methods called endga...

We use the method of pseudoanalytic continuation to obtain a characterization spaces holomorphic functions with boundary values in Besov terms polynomial approximations.

This paper is a continuation of paper [1] where we proved that for every linear finitely generated group G and any injective endomorphism φ of G, the mapping torus of φ is residually finite. The mapping torus of φ is the following ascending HNN extension of G: HNNφ (G) = 〈G, t | txt−1 = φ(x)〉 where x runs over a (finite) generating set of G. Probably, the most important mapping tori are mapping...

Homotopy continuation methods to compute numerical approximations to all isolated solutions of a polynomial system are known as “embarrassingly parallel”, i.e.: because of their low communication overhead, these methods scale very well for a large number of processors. Because so many important problems remain unsolved mainly due to their intrinsic computational complexity, it would be embarras...

This note is a continuation of a paper by the same authors that appeared in 2002 in the same journal. First we extend the method of the previous paper proving an asymptotic formula for the number of permutations for which the associated permutation polynomial has d coefficients in specified fixed positions equal to 0. This also applies to the function Nq,d that counts the number of permutations...