## “School of Mathematics”

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Paper   IPM / M / 17153
 School of Mathematics Title: Recursive structures in involutive bases theory Author(s): Amir Hasemi (Joint with M. Orth and W. M. Seiler) Status: Published Journal: J. Symb. Comput. Vol.: 118 Year: 2023 Pages: 32-68 Supported by: IPM
Abstract:
We study characterisations and corresponding completion algorithms of involutive bases using a recursion over the variables in the underlying polynomial ring. Key ingredients are an old result by Janet recursively characterising Janet bases for which we provide a new and simpler proof, the Berkesch--Schreyer variant of Buchberger's algorithm and a tree representation of set of terms also known as Janet trees. We start by extending 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 results to Pommaret bases. We give a novel recursive characterisation of quasi-stability and use it 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.