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Paper   IPM / M / 15680
School of Mathematics
  Title:   Degree upper bounds for involutive bases
  Author(s):  Amir Hashemi (Joint with H. Parnian and W. M. Seiler)
  Status:   Published
  Journal: Mathematics in Computer Science (MCS)
  No.:  2
  Vol.:  15
  Year:  2021
  Pages:   233-254
  Supported by:  IPM
The aim of this paper is to investigate upper bounds for the maximum degree of the elements of any minimal Janet basis of an ideal generated by a set of homogeneous polynomials. The presented bounds depend on the number of variables and the maximum degree of the generating set of the ideal. For this purpose, by giving a deeper analysis of the method due to Dube (SIAM J Comput 19:750–773, 1990), we improve (and correct) his bound on the degrees of the elements of a reduced Grobner basis. By giving a simple proof, it is shown that this new bound is valid for Pommaret bases, as well. Furthermore, based on Dube's method, and by introducing two new notions of genericity, so-called J-stable position and prime position, we show that Dube's (new) bound holds also for the maximum degree of polynomials in any minimal Janet basis of a homogeneous ideal in any of these positions. Finally, we study the introduced generic positions by proposing deterministic algorithms to transform any given homogeneous ideal into these positions.

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