Newton process and semigroups of irreducible quasi-ordinary power series

Abstract

The Newton process were introduced by Artal-Bartolo, Cassou-Noguès, Luengo and Melle-Hernández as a generalization of the Newton algorithm associated to plane curve singularities. Newton process is useful to study $\nu$-quasi-ordinary and quasi-ordinary polynomials in any number of variables. We describe numerically the Newton process associated to a quasi-ordinary branch of an irreducible quasi-ordinary polynomial in terms of its characteristic exponents. We show the relation between these numerical data and the semigroup of the singularity, give a criterium for irreducibility of quasi-ordinary polynomials and describe the normalization of irreducible quasi-ordinary surfaces in terms of the numerical data. We also study why and when irreducibility fails to be preserved by the Newton process.

Appendix: Newton trees

In this subsection we illustrate graphically a Newton process associated to a q.o. polynomial, compare this illustration with the Newton trees introduced in [3–5] and give an alternative explanation to the canonical equations of q.o. hypersurfaces with given characteristic exponents mentioned in Remark 2.12.

Firstly, we represent a Newton process associated to a q.o. polynomial $f\in \mathbf C \{x\}[z]$ with characteristic exponents $\lambda _j$ and characteristic integers $n_j$ with $1 \le j \le g$ by the following decorated graph (see Fig. 1). The decorations $(Q^{j}_1, \dots , Q^{j}_d)$ and $n_j$ are around the $j$th node but $(Q^{j}_1, \dots , Q^{j}_d)$ lies in the horizontal edge (in the upward vertical arrow for $j=1$) while $n_j$ lies in the (downward) vertical arrow. Omit for the moment the variables attached to the arrows.

Fig. 1

Decorated graph associated to the Newton process of an irreducible q.o. polynomial.

We use now a different notation for $s_j$ than those introduced in Notation 3.1. Let us denote $s_j$ as $(\frac{q^j_1}{n_j}, \dots , \frac{q^j _d}{n_j})$ where $n_j$ is the $j$th characteristic number and therefore we have that $\gcd (q^j_1, \dots , q^j_d, n_j) = r_j \ge 1$. Then we denote $(Q^{1}_1, \dots , Q^{1}_d) = n_1 s_1$ and

$$\begin{aligned} (Q^{j}_1, \dots , Q^{j}_d) = n_{j-1} n_j \left( \frac{Q^{j-1}_1}{\gcd (q^{j-1}_1, n_{j-1})}, \dots , \frac{Q^{j-1}_d}{\gcd (q^{j-1}_d, n_{j-1})}\right) + n_j s_j \end{aligned}$$

for $2 \le j \le g$. It is easy to compare the decorations with the elements appearing in proposition 4.7. Let us denote $\gcd (q^j, n_j)= (\gcd (q^j_1,n_1), \dots , \gcd (q^j_d, n_j))$ in particular we have that $(Q^{j}_1, \dots , Q^{j}_d)=\gcd (q^j, n_j) *\varvec{\gamma }^{[j]}_j$ for any $1 \le j \le g$. It is worthwhile to notice that this decorations can be defined from the inclinations $s_j$ by transformations rules similar to those of Proposition 4.7.

Remark 6.1

The decorated graph associated to the Newton process of an irreducible q.o. polynomial does not coincide with the Newton tree described in [3–5]. The difference arise when $f$ is false reducible at any iteration. If $f$ is false reducible at the $j$th iteration, the Newton tree is recovered from our diagram (and viceversa) unfolding the edge attached to the $j$th and $j+1$th nodes of our diagram into $r_j$ edges and dividing the decorations around the $j$th node by $r_j$. The Fig. 2 illustrates the difference for the polynomial of Example 3.6.

Fig. 2

Newton tree (left) and our decorated graph (right) for Example 3.6

Finally we describe how to obtain the relations (iv) of Lemma 2.10 from the combinatorics of the diagram. In the case of plane curves there exists a topological theory to explain why this combinatorics work [27].

Now we attach to the arrows of our diagram variables, as showed in Fig. 1. Following the language of Eisenbud and Neumann [12] and Neumann and Wahl [27] we define weights for the nodes of the diagram as the product of the decorations around the node; i.e. $\mathbf{w}(j) := n_j (Q^{j}_1, \dots , Q^{j}_d)$ for $j \in \{1,\dots ,g\}.$ We also define weights of the variables $x_1, \dots , x_d, z=u_1, \dots , u_{j+1}$ w.r.t. the $j$th nodes as the product of the normalized decorations adjacent to the path joining the node and the arrow corresponding to the variable. The adjective normalized in the definition of $\mathbf{w}(\cdot ,j)$ means that we divide the $i$th coordinate of the weight by $\gcd (n_r, q^r_i)$ if $r < j$. More precisely, we have the following relations $\mathbf{w}(x_i,j) = \gcd (q^j, n_j) *\varvec{\epsilon }^{[j]}_i$ for $1 \le i \le d$ and $1 \le j \le d$ and $\mathbf{w}(u_k,j) = \gcd (q^j, n_j) *{\varvec{\gamma }}^{[j]}_k$ for $1 \le k \le j$ and $1 \le j \le d$.

It is easy to check that the condition (iv) of Lemma 2.10 is equivalent to the existence of integers $0 \le l^{(j)}_i \le n_i-1$ and $\alpha _k^{(j)} \in \mathbf Z _{\ge 0}$ such that

$$\begin{aligned} \mathbf{w}(j) = \alpha ^{(j)}_1 \mathbf{w}(x_1,j) + \cdots + \alpha ^{(j)}_d \mathbf{w}(x_d,j) + l^{(j)}_1 \mathbf{w}(u_1,j) + \cdots + l^{(j)}_{j-1} \mathbf{w}(u_{j-1},j). \end{aligned}$$

Example 6.2

In the case of the irreducible q.o. polynomial of Example 4.8, we have that the decorations of the first node are $(Q^1_1, Q^1_2)=(2,5)$ and $n_1= 3$, of the second node are $(Q^2_1, Q^2_2)=(26,65)$ and $n_2= 4$ and of the third node are $(Q^3_1, Q^3_2)=(262, 1316)$ and $n_3= 5$. Moreover, we have that

$$\begin{aligned}&\mathbf{w}(x_1,1) = \varvec{\epsilon }^{[1]}_1 = (3,0), \quad \mathbf{w}(x_2,1) = \varvec{\epsilon }^{[1]}_2 = (0,3), \quad \mathbf{w}(u_1,1) = {\varvec{\gamma }}^{[1]}_1 = (2,5),\\&\mathbf{w}(x_1,2) = (12,0) \not = (6,0) = \varvec{\epsilon }^{[2]}_1, \quad \mathbf{w}(x_2,2) = \varvec{\epsilon }^{[2]}_2 = (0,12),\\&\mathbf{w}(u_1,2) = (8,20) \not = {\varvec{\gamma }}^{[2]}_1 = (4,20), \quad \mathbf{w}(u_2,2) = (26, 65) \not = {\varvec{\gamma }}^{[2]}_2 = (13, 65),\\&\mathbf{w}(x_1,3) = \varvec{\epsilon }^{[3]}_1 = (30,0), \quad \mathbf{w}(x_2,3) = \varvec{\epsilon }^{[3]}_2 = (0,60),\\&\mathbf{w}(u_1,3) = {\varvec{\gamma }}^{[3]}_1 = (20,100), \quad \mathbf{w}(u_2,3)= {\varvec{\gamma }}^{[3]}_1 = (65, 325), \\&\quad \mathbf{w}(u_3,3) = {\varvec{\gamma }}^{[3]}_1 = (262,1316). \end{aligned}$$