For a homogeneous polynomial of $n$ variables, we present a new method to compute the roots of Bernstein-Sato polynomial supported at the origin, assuming that general hyperplane sections of the associated projective hypersurface have at most weighted homogeneous isolated singularities. Calculating the dimensions of certain $E_r$-terms of the pole order spectral sequence for a given integer $r\in[2,n]$, we can detect its degeneration at $E_r$ for certain degrees. In the case of strongly free, locally positively weighted homogeneous divisors on ${\mathbb P}^3$, we can prove its degeneration almost at $E_2$ and completely at $E_3$ together with a symmetry of a modified pole-order spectrum for the $E_2$-term. These can be used to determine the roots of Bernstein-Sato polynomials supported at the origin, except for rather special cases.

中文翻译：

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某些具有二维奇异轨迹的齐次多项式的 Bernstein-Sato 多项式的根

对于 $n$ 变量的齐次多项式，我们提出了一种计算原点支持的 Bernstein-Sato 多项式的根的新方法，假设相关投影超曲面的一般超平面部分至多具有加权齐次孤立奇点。计算给定整数 $r\in[2,n]$ 的极序谱序列的某些 $E_r$ 项的维度，我们可以检测到它在 $E_r$ 处的退化程度。在 ${\mathbb P}^3$ 上的强自由、局部正加权齐次约数的情况下，我们可以证明它的退化几乎在 $E_2$ 和完全在 $E_3$ 以及修正的极阶谱的对称性对于 $E_2$-term。这些可用于确定在原点支持的 Bernstein-Sato 多项式的根，但相当特殊的情况除外。