关于平面分支的解析不变量和半根
On the Analytic Invariants and Semiroots of Plane Branches
摘要：
平面分支$C$的$k$ -半根$C_k$的值半群允许我们恢复$C$的值半群$Γ=\langle v_0,\ldots ,v_g\rangle$的一部分，即它与$C$的拓扑不变量有关。在本文中，我们考虑的一组值的微分$Λ_k$的$C_k$，这是一个分析不变量，我们展示了它如何确定的一组值的微分$Λ$的$C$。因此，在一个固定的拓扑类中，我们将$C$的Tjurina数$τ$与$C_k$的Tjurina数联系起来。特别地，我们证明了$τ\leq μ-\frac{3n_g-2}{4}μ_{g-1}$，其中$n_g=gcd(v_0,\ldots ,v_{g-1})$，$μ$和$μ_{g-1}$分别表示$C$和$C_{g-1}$的Milnor数.如果是$n_g=2$，则对于由$Γ$确定的拓扑类中的任何曲线，我们都有$τ=μ-μ_{g-1}$，这是Luengo和Pfister所得结果的推广。
英文摘要：
The value semigroup of a $k$-semiroot $C_k$ of a plane branch $C$ allow us to recover part of the value semigroup $Γ=\langle v_0,\ldots ,v_g\rangle$ of $C$, that is, it is related to topological invariants of $C$. In this paper we consider the set of values of differentials $Λ_k$ of $C_k$, that is an analytical invariant, and we show how it determine part of the set of values of differentials $Λ$ of $C$. As a consequence, in a fixed topological class, we relate the Tjurina number $τ$ of $C$ with the Tjurina number of $C_k$. In particular, we show that $τ\leq μ-\frac{3n_g-2}{4}μ_{g-1}$ where $n_g=gcd(v_0,\ldots ,v_{g-1})$, $μ$ and $μ_{g-1}$ denote the Milnor number of $C$ and $C_{g-1}$ respectively. If $n_g=2$, we have that $τ=μ-μ_{g-1}$ for any curve in the topological class determined by $Γ$ that is a generalization of a result obtained by Luengo and Pfister.
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