A short contribution to the theory of regular chains (Q2035618)

A regular chain is a triangular set of polynomials that satisfy some regularity assumptions. This paper builds on [\textit{F. Boulier} and \textit{F. Lemaire}, Math. Comput. Sci. 4, No. 2--3, 185--201 (2010; Zbl 1218.68202)] as well as [\textit{F. Boulier} et al., J. Symb. Comput. 93, 34--55 (2019; Zbl 1422.13023)] and improves them in some aspects. It presents (a) new algorithms for computing normal forms of rational fractions modulo a regular chain that will always succeed whenever a normal form exists and (b) a new proof for a membership test to an ideal defined by a regular chain. Algorithms and proof are conceptually simpler than existing versions and apply to both, the differential and the non-differential case.