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arXiv · 1405.0472

Border Bases for Polynomial Rings over Noetherian Rings

Ambedkar Dukkipati·Nithish Pai·Maria Francis
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Abstract

The theory of border bases for zero-dimensional ideals has attracted several researchers in symbolic computation due to their numerical stability and mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014), one can extend the concept of border bases over Noetherian rings whenever the corresponding residue class ring is finitely generated and free. In this paper we address the following problem: Can the concept of border basis over Noetherian rings exists for ideals when the corresponding residue class rings are finitely generated but need not necessarily be free modules? We present a border division algorithm and prove the termination of the algorithm for a special class of border bases. We show the existence of such border bases over Noetherian rings and present some characterizations in this regard. We also show that certain reduced Gr\"{o}bner bases over Noetherian rings are contained in this class of border bases.

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Ambedkar Dukkipati, Nithish Pai, Maria Francis. 2014-05-02. Border Bases for Polynomial Rings over Noetherian Rings. https://arxiv.org/abs/1405.0472

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