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We extend the concept of GrÃ¶bner bases to relative GrÃ¶bner bases for ideals in and modules over quotient rings of a polynomial ring over a field. We develop a ârelativeâ variant of both Buchbergerâs criteria for avoiding reductions to zero and Schreyerâs theorem for a GrÃ¶bner basis of the syzygy module. As main contribution, we then introduce the novel notion of relative involutive bases and present an algorithm for their explicit construction. Finally, we define the new notion of relatively quasi-stable ideals and exploit it for the algorithmic determination of coordinates in which finite relative Pommaret bases exist.
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