Computation of inverses in residue class rings of parametric polynomial ideals

For a given polynomial <i>f</i> and an ideal <i>I</i> of a polynomial ring <i>K</i>[<i>X</i>] over a field <i>K</i>, we give a necessary and sufficient condition for <i>I</i> to have a smallest ideal extension <i>J</i> such that <i>f</i> is invertible in the residue class ring <i>K</i>[<i>X</i>]/<i>J</i>. If the condition holds, <i>J</i> is shown to be the saturation ideal <i>I</i>:∞. We also show primary decompositions of the ideals <i>I</i>:∞ and <i>I</i>+9<i>f</i><sup><i>m</i></sup>;:, where <i>m</i> is a natural number such that <i>I</i>:∞ = <i>I</i>:<i>f</i><sup><i>m</i></sup>, all together forms a primary decomposition of <i>I</i> with no modification. These observations are especially useful for polynomial rings with parameters.

[1]  Kazuhiro Yokoyama.  Stability of Parametric Decomposition , 2006, ICMS.

[2]  佐藤 洋祐,et al.  特集 Comprehensive Grobner Bases , 2007 .

[3]  Katsusuke Nabeshima,et al.  On the computation of parametric Gröbner bases for modules and syzygies , 2010 .

[4]  Antonio Montes,et al.  A New Algorithm for Discussing Gröbner Bases with Parameters , 2002, J. Symb. Comput..

[5]  Antonio Montes,et al.  Improving the DISPGB algorithm using the discriminant ideal , 2006, J. Symb. Comput..

[6]  Masayuki Noro,et al.  A Computer Algebra System: Risa/Asir , 2003, Algebra, Geometry, and Software Systems.

[7]  Volker Weispfenning,et al.  Comprehensive Gröbner Bases , 1992, J. Symb. Comput..

[8]  Akira Suzuki,et al.  An alternative approach to comprehensive Gröbner bases , 2002, ISSAC '02.

[9]  Akira Suzuki,et al.  A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases , 2006, ISSAC '06.

[10]  Deepak Kapur,et al.  An Approach for Solving Systems of Parametric Polynomial Equations , 1993 .

[11]  Katsusuke Nabeshima,et al.  A speed-up of the algorithm for computing comprehensive Gröbner systems , 2007, ISSAC '07.

[12]  Dominique Duval,et al.  Algebraic Numbers: An Example of Dynamic Evaluation , 1994, J. Symb. Comput..

[13]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[14]  임종인,et al.  Gröbner Bases와 응용 , 1995 .

[15]  Masayuki Noro.  Modular dynamic evaluation , 2006, ISSAC '06.