Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension One

Abstract

Abstract. For a regular chain R in dimension one, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of R, that is, the set W (R) \ W (R). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of R. We provide experimental results illustrating the benefits of our algorithms.

Regular Chains under Linear Changes of Coordinates and Applications

“…Moreover, these new techniques can handle cases where the results of our previous paper [1] could not apply. One of the main ideas of our new results (see for instance Theorem 2) is to use a linear change of coordinates so as to replace the description of W (T ) by one for which W (T ) ∩ V (h T ) can be computed by means of standard operations on regular chains.…”

“…Moreover, the base field k is either R or C so that the affine space k n is endowed with the Euclidean topology. In this context, we recall from [1] that the quasicomponent W (T ) has the same closure in both the Euclidean and the Zariski topologies.…”

“…where lim(W (T )) := W (T ) ∩ V (h T ) holds and the points of that latter set are called the (non-trivial) limit points of W (T ), for reasons explained in [1]. We say that T is a regular chain whenever T is empty or T <w is a regular chain and the initial of T w is regular modulo sat(T <w ), where w is the largest main variable of a polynomial in …”

“…Our techniques (see Proposition 2, Theorem 2, Theorem 3, Theorem 4 and Algorithm 1) rely on linear changes of coordinates and allow us to relax the "dimension one" hypothesis in our previous paper [1], where lim(W (T )) was computed via Puiseux series.…”

Doing algebraic geometry with the RegularChains library

“…In [1], three of the co-authors of this note have recently proposed a procedure for computing the non-trivial limit points of the quasi-component W (R), that is, the set lim(W (R)) := W (R) \ W (R) as a finite union of quasi-components of some other regular chains. This procedure, currently implemented in the case where sat(R) has dimension one, relies only on operations on regular chains, like the regularity test.…”