利用半连续性进行标准基计算
Using Semicontinuity for Standard Bases Computations
AI总结：
本文提出利用半连续性理论加速标准基计算，通过选择素数或常数来减少系数膨胀，提升计算效率。
摘要：
我们提出了新的结果和一个算法，用于计算一个0维理想I在幂级数环或在有限变量多项式环的局部化中的标准基。该算法在K是Noetherian积分域A的商域时，当系数膨胀发生时，能提供显著的加速效果。最重要的特殊情况可能是当A是整数环或当A是某些有限参数的多项式环上时。给定I作为A上的多项式环中的理想，我们首先计算一个模素数p的标准基，或通过专门化参数到常数来计算。然后利用专门化理想中的“最高角落”来在标准基计算过程中截断多项式中的高次项，从而实现加速。一个重要事实是，我们可以选择p为任意素数或任意常数，而不只是“幸运”或“随机”的一个。算法的正确性将从第一作者和第二作者的通用半连续性定理中推导出来。计算机代数系统Singular已经实现了该算法的功能，我们提供了几个例子来说明其威力。
英文摘要：
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.
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