关于FGLM算法的p-adic稳定性
On the p-adic stability of the FGLM algorithm
摘要：
目前，许多求解多项式系统的策略都是先计算Gr{<$}bner基，然后再通过变序算法得到Gr{<$}bner基。排序算法的改变是这些策略的关键。研究了主要的变序算法FGLM的p-adic稳定性。我们证明了FGLM是稳定的，并给出了其执行过程中精度损失的上界。FGLM的变体被设计成在形状位置上从grevlex排序传递到Gr{<$}bner基也是稳定的。我们的研究依赖于Smith范式计算在线性代数中的应用。
英文摘要：
Nowadays, many strategies to solve polynomial systems use the computation of a Gr{ö}bner basis for the graded reverse lexicographical ordering, followed by a change of ordering algorithm to obtain a Gr{ö}bner basis for the lexicographical ordering. The change of ordering algorithm is crucial for these strategies. We study the p-adic stability of the main change of ordering algorithm, FGLM. We show that FGLM is stable and give explicit upper bound on the loss of precision occuring in its execution. The variant of FGLM designed to pass from the grevlex ordering to a Gr{ö}bner basis in shape position is also stable. Our study relies on the application of Smith Normal Form computations for linear algebra.
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