Using Semicontinuity for Standard Bases Computations
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arXiv
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| Format: | Preprint |
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2021
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| author | Greuel, Gert-Martin Pfister, Gerhard Schönemann, Hans |
| author_facet | Greuel, Gert-Martin Pfister, Gerhard Schönemann, Hans |
| contents | We present new results and an algorithm for standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring in finitely many variables over a field K. The algorithm provides a significant speed up if K is the quotient field of a Noetherian integral domain A, when coefficient swell occurs. The most important special cases are perhaps when A is the ring of integers resp. when A is a polynomial ring over some field in finitely many parameters. Given I as an ideal in the polynomial ring over A, we compute first a standard basis modulo a prime number p, resp. by specializing the parameter to a constant. We then use the "highest corner" of the specialized ideal to cut off high order terms from the polynomials during the standard basis computation over K to get the speed up. An important fact is that we can choose p as an arbitrary prime resp. as an arbitrary constant, not just a "lucky" resp. "random" one. Correctness of the algorithm will be deduced from a general semicontinuity theorem due to the first two authors. The computer algebra system Singular provides already the functionality to realize the algorithm and we present several examples illustrating its power. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_09735 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Using Semicontinuity for Standard Bases Computations Greuel, Gert-Martin Pfister, Gerhard Schönemann, Hans Commutative Algebra Algebraic Geometry 1304, 13P10, 1404, 14B05, 14Q20 We present new results and an algorithm for standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring in finitely many variables over a field K. The algorithm provides a significant speed up if K is the quotient field of a Noetherian integral domain A, when coefficient swell occurs. The most important special cases are perhaps when A is the ring of integers resp. when A is a polynomial ring over some field in finitely many parameters. Given I as an ideal in the polynomial ring over A, we compute first a standard basis modulo a prime number p, resp. by specializing the parameter to a constant. We then use the "highest corner" of the specialized ideal to cut off high order terms from the polynomials during the standard basis computation over K to get the speed up. An important fact is that we can choose p as an arbitrary prime resp. as an arbitrary constant, not just a "lucky" resp. "random" one. Correctness of the algorithm will be deduced from a general semicontinuity theorem due to the first two authors. The computer algebra system Singular provides already the functionality to realize the algorithm and we present several examples illustrating its power. |
| title | Using Semicontinuity for Standard Bases Computations |
| topic | Commutative Algebra Algebraic Geometry 1304, 13P10, 1404, 14B05, 14Q20 |
| url | https://arxiv.org/abs/2108.09735 |