Symbolic Hamburger-Noether expressions

We recall the definition of Hamburger-Noether expansions (HNE), resp. symbolic Hamburger-Noether expressions, for the branches of a reduced plane curve singularity. They can be regarded as the analogue of Puiseux expansions when working over a field of positive characteristic (cf. the discussion in [2], 2.1). As being the case for Puiseux expansions, many invariants of plane curve singularities (such as the multiplicity sequence, the $\delta$-invariant, the intersection multiplicities of the branches, etc.), can be computed directly from the corresponding system of HNE.

Let $P\in C$ be a point and $x,y$ local parameters at $P$. Moreover, let the germ $(C,P)$ be given by a local equation $f\in \mathbf{F}[[x,y]]$ with irreducible decomposition $f=f_1\cdot\ldots\cdot f_r\in \overline{\mathbf{F}}[[x,y]]$. Finally, let's suppose that $x$ is a transversal parameter for $(C,P)$, that is, the order of $f(0,y)$ is equal to the order of $f(x,y)$.

Definition 2.1   A Hamburger-Noether expansion (HNE) of $C$ at $P$ for the local branch given by $f_\nu$ (defined over some finite algebraic extension $\mathbf{F}\subset \mathbf{F}_{\text{ext}}$) is a finite sequence of equations

$$z_{-1} = a_{0,1}z_0 + a_{0,2}z_0^2 + \ldots + a_{0,h_0}z_0^{h_0}\! +z_0^{h_0}z_1$$

$$z_{0} = \phantom{a_{0,1}z_0 + }\;\, a_{1,2}z_1^2 + \ldots + a_{1,h_1}z_1^{h_1}\! +z_1^{h_1}z_2$$

$$\vdots\phantom{i} \phantom{a_{0,1}z_0 + A} \vdots$$

$$z_{i-1} = \phantom{a_{0,1}z_0 + }\;\, a_{i,2}z_i^2 + \ldots + a_{i,h_i}z_i^{h_i}\! +z_i^{h_i}z_{i+1}$$ (1)

$$\vdots\phantom{i} \phantom{a_{0,1}z_0 + A} \vdots$$

$$z_{s-2} = \phantom{a_{0,1}z_0 + }\;\, a_{s-1,2}z_{s-1}^2 + \ldots + a_{s-1,h_{s-1}}z_{s-1}^{h_{s-1}}\! +z_{s-1}^{h_{s-1}}z_{s}$$

$$z_{s-1} = \phantom{a_{0,1}z_0 + }\;\, a_{s,2}z_{s}^2 + a_{s,3}z_{s}^3 + \ldots \ldots \ldots \ldots$$

where $s$ is a non-negative integer, $a_{j,i}\in \mathbf{F}_{\text{ext}}$ and $h_j$, $j=1,\dots ,s-1$, are positive integers, such that $f_\nu\bigl(z_0(t),z_{-1}(t)\bigr) = 0$    in $\mathbf{F}_{\text{ext}}[[z_s]].$

If the local equation of $(C,P)$ is polynomial in $x,y$, i.e., $f\in \mathbf{F}[x,y]$, then the last (infinite) row of (1) can be replaced, equivalently, by a (finite) implicit equation

$$g(z_s,z_{s-1})\,=\,0\,,\qquad g\in \mathbf{F}_{\text{ext}}[x,y]\,, \quad \frac{\partial g}{\partial z_{s-1}} (0,0) \neq 0\,.$$

The resulting system is called a symbolic Hamburger-Noether expression (sHNE) for the branch.

Any HNE leads to a primitive parametrization $\varphi: \mathbf{F}[[x,y]]\to \mathbf{F}_{\text{ext}}[[t]]$ of the branch (setting $t:=z_s$ and mapping $x\mapsto z_0(z_s)$, $y\mapsto z_{-1}(z_s)$). It can be computed from a sHNE up to an arbitrary finite degree in $t$.

Remark 2.2   There exist constructive algorithms to compute a system of sHNE's (resp. HNE's up to a given degree) for the branches of a reduced plane curve singularity (cf. [2] for the irreducible case, resp. [34] for the reducible case). A modification of the latter algorithm is implemented in the computer algebra system SINGULAR [26,17].

To perform the algorithm one does not need any knowledge about the irreducible factorization of $f$ in $\overline{\mathbf{F}}[[x,y]]$. Moreover, in the reducible case, the probably necessary (finite) field extension is not performed a priori, but by successive (primitive) extensions introduced exactly when needed for computations (factorization).

Remark 2.3   Given a Hamburger-Noether expansion (1) for a local branch $(C_\nu,P)$, the corresponding multiplicity sequence $m_1,\dots,m_n$ can be easily read off:

$$\underbrace{n_0,\dots,n_0}_{h_0\text{ times}}, \underbrace{n_1,\... ...ots\ldots, \underbrace{n_{s-1},\dots,n_{s-1}}_{h_{s-1}\text{ times}},1,\dots,1$$

where the $n_j$ are defined recursively by setting $n_s:=1$ and, for $j=s,\dots,1$,

$$n_{j-1} := \left\{ \begin{array}{ll} n_jh_j\!+n_{j+1} & \text... ...ext{ minimal s.th.\ $a_{j,\ell}\neq 0$}. \end{array}\right.$$

In particular, we can compute the $\delta$-invariant of the branch directly from the Hamburger-Noether expansion, since

$$\delta(C_\nu,P)=\sum_{i=1}^n \frac{m_i(m_i\!-1)}{2}\,.$$

Moreover, if $\varphi: \mathbf{F}[[x,y]]\to \mathbf{F}_{\text{ext}}[[t]]$ is the primitive parametrization of $(C_\nu,P)$ obtained from the HNE then the intersection multiplicity of $(C_\nu,P)$ with the plane curve germ $(C'\!,P)$ given by $g\in \overline{\mathbf{F}}[[x,y]]$ can be computed as $i_P(C_\nu,C')= ord_t g\bigl(\varphi(x),\varphi(y)\bigr)$.