Computation of the Minimal Associated Primes-猫眼课题宝

最小关联素数的计算

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2006

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Santiago Laplagne

求解多项式方程组是计算机代数中的一项主要任务，尽管什么是可接受的解决方案的确切含义取决于上下文。在这次演讲中，我们将其解释为寻找由多项式生成的理想的最小相关素数。在几何上，这等价于将解集分解成其不可约分量。我们研究了现有的算法，并提出了一些修改。一个常用的技术是减少问题的零维情况。在Gianni、Trager和Zacharias的一篇论文中，他们使用了这种技术，结合分裂工具$I =（I：h^infty）cap langle I，h^m 对于一些特定的多项式$h$和整数$m$。这种分裂引入了许多不属于原始理想的冗余组件。在我们这里提出的算法中，我们使用了零维情况的简化，但我们避免了使用理想的$langle I，h^m 角$。因此，当理想具有不同维度的组件时，我们的算法通常更有效。

Solving systems of polynomial equations is a main task in Computer Algebra, although the precise meaning of what is an acceptable solution depends on the context. In this talk, we interpret it as finding the minimal associated primes of the ideal generated by the polynomials. Geometrically, this is equivalent to decompose the set of solutions into its irreducible components. We study the existing algorithms, and propose some modifications. A common technique used is to reduce the problem to the zero dimensional case. In a paper by Gianni, Trager and Zacharias they use this technique, combined with the splitting tool $I = (I : h^infty) cap langle I, h^m angle$ for some specific polynomial $h$ and integer $m$. This splitting introduces a number of redundant components that are not part of the original ideal. In the algorithm we present here, we use the reduction to the zero dimensional case, but we avoid working with the ideal $langle I, h^m angle$. As a result, when the ideal has components of different dimensions, our algorithm is usually more efficient.

Computation of the Minimal Associated Primes

最小关联素数的计算

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