Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension One Parisa Alvandi, Changbo Chen, Marc Moreno Maza ORCCA, University of Western Ontario (UWO), London, Ontario, Canada palvandi@uwo.ca, changbo.chen@gmail.com, moreno@csd.uwo.ca Abstract. For a regular chain R in dimension one, we propose an algo- rithm 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 se- ries 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. 1 Introduction The theory of regular chains, since its introduction by J.F. Ritt [22], has been applied successfully in many areas including differential systems [8, 2, 13], differ- ence systems [12], unmixed decompositions [14] and primary decomposition [23] of polynomial ideals, intersection multiplicity calculations [17], cylindrical alge- braic decomposition [7], parametric [28] and non-parametric [4] semi-algebraic systems. Today, regular chains are at the core of algorithms computing triangular decomposition of polynomial systems and which are available in several software packages [16, 26, 27]. Moreover, those algorithms provide back-engines for com- puter algebra system front-end solvers, such as Maple’s solve command. This paper deals with a notorious issue raised in all types of triangular decom- positions, the Ritt problem, stated as follows. Given two regular chains (algebraic or differential) R and S, whose saturated ideals sat(R) and sat(S) are radical, check whether the inclusion sat(R) ⊆ sat(S) holds or not. In the algebraic case, the challenge is to test such inclusion without computing a system of genera- tors of sat(R). This question would be answered if one would have a procedure with the following specification: for the regular chain R compute regular chains R 1 ,...,R e such that W (R)= W (R 1 ) ∪···∪ W (R e ) holds, where W (R) is the quasi-component of R and W (R) is the Zariski closure of W (R). We propose a solution to this algorithmic quest, in the algebraic case. To be precise, our procedure computes 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, see Theorem 7 in Section 7. We focus on the case where sat(R) has dimension one. When the regular chain R consists of a single polynomial r, primitive w.r.t. its main variable, one can easily check that lim(W (R)) = V (r, h r ) holds, where