齐次理想的Bernstein-Sato多项式的$-n/d$根的多项式PDE准则
A Polynomial PDE Criterion for the $-n/d$ Root of the Bernstein-Sato polynomial of Homogeneous Ideals
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对由次数为d的齐次多项式生成的理想，给出判定其Bernstein-Sato多项式$-n/d$根的多项式PDE准则，应用于极大子式理想得特殊根$-n$，且该准则具稳定性。
AI中文摘要：
设$I\subseteq\mathbb C[x_1,\ldots,x_n]$是由次数均为$d$的齐次多项式生成的理想。我们给出一个多项式偏微分方程准则，用于判定$-n/d$是否为Bernstein-Sato多项式$b_I(s)$的根。该准则通过研究与$I$的生成元相关联的图$\mathcal D$-模的一个商，再借助Fischer配对对偶化一个辅助有限多项式PDE系统得到。我们将该准则应用于一般$m\times n$矩阵的极大子式理想，得到特殊根$-n$；结合沿行列式层的局部可除性，这在极大子式情形下得到强单值性结论。最后，我们证明该准则在以下操作下具有稳定性：扩大生成元的线性张成、在不相交变量中添加生成元、满足自然斜率条件的乘积，以及Thom-Sebastiani和。这些稳定性结果提供了新的齐次理想与多项式类，无需计算完整的Bernstein-Sato多项式即可检测其特殊Bernstein-Sato根。关键词：Bernstein-Sato多项式，单值性猜想。
英文摘要：
Let $I\subseteq\mathbb C[x_1,\ldots,x_n]$ be an ideal generated by homogeneous polynomials of a common degree $d$. We give a polynomial partial differential equation criterion guaranteeing that $-n/d$ is a root of the Bernstein-Sato polynomial $b_I(s)$. We apply this criterion to the ideal of maximal minors of a generic $m\times n$ matrix and obtain the distinguished root $-n$; combined with local divisibility along determinantal strata, this allows us to obtain the strong monodromy conjecture in the maximal-minor case. Finally, we prove that the criterion is stable under enlarging the linear span of the generators, adjoining generators in disjoint variables, products satisfying the natural slope condition, and Thom-Sebastiani sums. These stability results provide new classes of homogeneous ideals and polynomials for which the distinguished Bernstein-Sato root can be detected. Keywords. Bernstein-Sato polynomial, monodromy conjecture.
