AI 中文总结
研究布伦特方程解的各向同性群作用问题，通过固定部分解的方法使参数化解集与不同轨道相交，从已知解获得非平凡参数化解集，如杜马斯等人找到的解被参数化，还得到无穷多个仅含有理系数的48次乘法的不等价算法。
AI 中文摘要
给定布伦特方程的一个解，我们通常固定其一个部分解并进行代换。在约化多项式系统中能找到一个参数化解集，但解集中存在正维各向同性群作用，参数化解可能属于同一轨道。本文找到一种固定部分解的方法，使参数化解集与不同轨道相交，可从已知解得到非平凡参数化解集。比如杜马斯等人找到的解被参数化，在该参数化解集中能找到无穷多个仅含有理系数的48次乘法的不等价算法。
英文摘要
Given a solution of the Brent equations, a partial solution can be substituted into the Brent equations which makes it a reduced polynomial system. Then the reduced polynomial system can be solved by symbolic software, and a parameterized solution set can often be found. However, the solution set admits a positive-dimensional isotropy group action. The parameterized solution set may lie entirely in a single isotropy group orbit. In this paper, we give a geometric explanation of making the substitution. Considering the foliation structure of the solution set, a method on choosing the partial solution is proposed. By the proposed method, parameterized solution sets that intersect distinct isotropy group orbits can be efficiently determined. In particular, the rational solution found by Dumas, Pernet and Sedoglavic is parameterized. In the parameterized solution set, we can find infinitely many inequivalent 48-multiplication algorithms with only rational coefficients.
CommentsThe second version, many typos have been changed
