Recursive structures in involutive bases theory

We study characterisations of involutive bases using a recursion over the variables in the underlying polynomial ring and corre-sponding completion algorithms. Three key ingredients are (i) an old result by Janet recursively characterising Janet bases for which we provide a new and simpler proof, (ii) the Berkesch-Schreyer variant of Buchberger's algorithm and (iii) a tree representation of sets of terms also known as Janet trees. We start by extend-ing Janet's result to a recursive criterion for minimal Janet bases leading to an algorithm to minimise any given Janet basis. We then extend Janet's result also to Janet-like bases as introduced by Gerdt and Blinkov. Next, we design a novel recursive completion algorithm for Janet bases. We study then the extension of these re-sults to Pommaret bases. It yields a novel recursive characterisation of quasi-stability which we use for deterministically constructing "good" coordinates more efficiently than in previous works. A small modification leads to a novel deterministic algorithm for putting an ideal into N oe ther position. Finally, we provide a general theory of involutive-like bases with special emphasis on Pommaret-like bases and study the syzygy theory of Janet-like and Pommaret-like bases.(c) 2023 Elsevier Ltd. All rights reserved.