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Paper IPM / M / 17153  


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 BerkeschSchreyer 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 Janetlike 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 quasistability 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 involutivelike
bases with special emphasis on Pommaretlike bases and study the syzygy
theory of Janetlike and Pommaretlike bases.
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