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Current Challenges in Developing Open Source Computer Algebra Systems Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Solving Extended Ideal Membership Problems in Rings of Convergent Power Series via Gröbner Bases

Authors : Katsusuke Nabeshima, Shinichi Tajima

Published in: Mathematical Aspects of Computer and Information Sciences

Publisher: Springer International Publishing

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Abstract

An extended ideal membership algorithm is considered in the ring of convergent power series. It is shown that the problem for zero-dimensional ideals in a local ring can be solved in a polynomial ring. The key of the proposed method is the use of ideal quotients in polynomial rings. A new algorithm is given to solve the extended ideal membership problems in local rings. A generalization of the resulting algorithm to ideals with parameters is also described.

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previous chapter Efficient Subformula Orders for Real Quantifier Elimination of Non-prenex Formulas
next chapter Advanced Algebraic Attack on Trivium
Footnotes
1
The degree reverse lex. monomial order with the coordinate (x, y) or (x, y, z), is used in the implementation of ExtIMP.
 
2
syz \((g_1,g_2,\ldots ,g_r)\) outputs a standard basis of the module of syzygies w.r.t. the generators \(g_1,g_2,\ldots ,g_r\) where \(g_1,g_2,\ldots ,g_r \in \mathbb {Q}[x]\). Thus, the command syz outputs the similar results. For each \(i \in \{1,\ldots , 8\}\), syz \((h,\frac{\partial f_i}{\partial x},\frac{\partial f_i}{\partial y},\frac{\partial f_i}{\partial z})\) (or syz \((h,\frac{\partial f_i}{\partial x},\frac{\partial f_i}{\partial y})\)) has been executed in Table 1.
 
3
The negative degree reverse lex. monomial order with the coordinate (x, y) or (x, y, z), is used in Singular’s command syz.
 
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Metadata
Title
Solving Extended Ideal Membership Problems in Rings of Convergent Power Series via Gröbner Bases
Authors
Katsusuke Nabeshima
Shinichi Tajima
Copyright Year
2016
Book
Mathematical Aspects of Computer and Information Sciences

Print ISBN: 978-3-319-32858-4

Electronic ISBN: 978-3-319-32859-1

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-32859-1_22

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