Computer-assisted proofs for finding the monodromy of Picard-Fuchs differential equations for a family of K3 toric hypersurfaces
Toshimasa Ishige, Akitoshi Takayasu
在本文中,我们提出了一种数值方法,用于严格地查找线性微分方程的单数。 从由串流明确给出某些特定解决方案的基础点开始,我们首先使用区间算术计算基本解决方案系统的价值,以严格控制截断和四舍五入错误。 然后,通过严格的集成器,沿着规定的轮廓分析继续解法,包围微分方程的奇异点。 从这些计算中,导出单体矩阵,生成微分方程的单体组。 这种方法建立了一个数学上严格的框架来解决微分方程中的单数问题。 对于一个值得注意的例子,我们应用我们的计算机辅助证明方法来解决与K3 toric hypersurfaces家族相关的Picard-Fuchs微分方程的单数问题。
In this paper, we present a numerical method for rigorously finding the monodromy of linear differential equations. Beginning at a base point where certain particular solutions are explicitly given by series expansions, we first compute the value of fundamental system of solutions using interval arithmetic to rigorously control truncation and rounding errors. The solutions are then analytically continued along a prescribed contour encircling the singular points of the differential equation via a...
