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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
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.

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