论文标题
zeta功能和$μ^*$ - Zariski对表面
Zeta-function and $μ^*$-Zariski pairs of surfaces
论文作者
论文摘要
一对Zariski表面是$ \ Mathbb {C}^3 $中的一对复杂的多项式函数，它是从经典的Zariski对Zariski对一对投射曲线$ f_0（z_1，z__2，z_3，z_3）= 0 $ f_1和$ f_1和$ f_1（z_1，z_1，z_2，z_2，z_3）= 0 $ a的$ f_1 = 0 $ dy $ dy $ d $ a same term of the form $z_i^{d+m}$ ($m\geq 1$) to both $f_0$ and $f_1$ so that the corresponding affine surfaces of $\mathbb{C}^3$ -- defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ -- have与其Milnor纤维相关的单片的孤立奇异性和相同的Zeta功能（尤其是$ g_0 $和$ g_1 $具有相同的milnor编号）。在本文中，我们表明，如果$ f_0 $和$ f_1 $相对于坐标$（z_1，z_2，z_3）$“方便”，并且如果曲线的奇异性$ f_0 = 0 $ f_1 = 0 $ and $ f_1 = 0 $ newton newton newton newton newton newton newton in Newton newton in Newton newton in Newton newtor in Newton newtrese n newtor Anew newto n newter Anew newtore in Pocal coordinates in Bocornate $（ZAR $（ZAR）$（ZAR）$（G_1）$（g_1）表面，即一对Zariski对多项式$ g_0 $和$ g_1 $的表面，具有相同的Teissier的$μ^*$ - 序列，但位于$μ^*$的不同路径连接组件中 - 恒定稳定性。为此，我们证明了一个新的通用公式，在适当的条件下，在适当的条件下，上述类型的函数数量，我们（在一般环境中）表明，两个多项式函数位于$μ^*$的相同路径连接组件中 - 恒定阶层 - 始终可以通过“零件复杂的分析路径”连接。
A Zariski pair of surfaces is a pair of complex polynomial functions in $\mathbb{C}^3$ which is obtained from a classical Zariski pair of projective curves $f_0(z_1,z_2,z_3)=0$ and $f_1(z_1,z_2,z_3)=0$ of degree $d$ in $\mathbb{P}^2$ by adding a same term of the form $z_i^{d+m}$ ($m\geq 1$) to both $f_0$ and $f_1$ so that the corresponding affine surfaces of $\mathbb{C}^3$ -- defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ -- have an isolated singularity at the origin and the same zeta-function for the monodromy associated with their Milnor fibrations (so, in particular, $g_0$ and $g_1$ have the same Milnor number). In the present paper, we show that if $f_0$ and $f_1$ are "convenient" with respect to the coordinates $(z_1,z_2,z_3)$ and if the singularities of the curves $f_0=0$ and $f_1=0$ are Newton non-degenerate in some suitable local coordinates, then $(g_0,g_1)$ is a $μ^*$-Zariski pair of surfaces, that is, a Zariski pair of surfaces whose polynomials $g_0$ and $g_1$ have the same Teissier's $μ^*$-sequence but lie in different path-connected components of the $μ^*$-constant stratum. To this end, we prove a new general formula that gives, under appropriate conditions, the Milnor number of functions of the above type, and we show (in a general setting) that two polynomials functions lying in the same path-connected component of the $μ^*$-constant stratum can always be joined by a "piecewise complex-analytic path".
