“School of Mathematics”
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| Paper IPM / M / 17153 |
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| Abstract: | |
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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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