Border Bases for Polynomial Rings over Noetherian Rings
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.
Cite
@article{arxiv.1405.0472,
title = {Border Bases for Polynomial Rings over Noetherian Rings},
author = {Ambedkar Dukkipati and Nithish Pai and Maria Francis},
journal= {arXiv preprint arXiv:1405.0472},
year = {2017}
}