Jumps of Milnor numbers of Brieskorn-Pham singularities in non-degenerate families

TL;DR

This paper provides an explicit formula and an inductive algorithm to compute the minimal non-zero change in Milnor numbers for non-degenerate deformations of certain singularities, specifically Brieskorn-Pham types.

Contribution

It introduces a novel inductive algorithm based on Diophantine equations to calculate the non-degenerate jump of Milnor numbers for specific singularities.

Findings

Derived a formula for the non-degenerate jump of Milnor numbers.

Developed an inductive algorithm using Diophantine equations.

Applied the method to Brieskorn-Pham singularities.

Abstract

The jump of the Milnor number of an isolated singularity $f_{0}$ is the minimal non-zero difference between the Milnor numbers of $f_{0}$ and one of its deformation $(f_{s}).$ In the case $f_{s}$ are non-degenerate singularities we call the jump non-degenerate. We give a formula (an inductive algorithm using diophantine equations) for the non-degenerate jump of $f_{0}$ in the case $f_{0}$ is a convenient singularity with only one $(n-1)$-dimensional face of its Newton diagram which equivalently (in our problem) can be replaced by the Brieskorn-Pham singularities.

Figures

Equations

$${\text{upmu}}_s:=\text{upmu}(f_s)={\dim }_{\text{mathbbm}C}{\mathcal{O}}_n/(\nabla f_s)\text{missing},$$

$$f_0(\text{z})=z_1^{p_1}+\dots +z_n^{p_n},p_i⩾2,i=1,\dots ,n,$$

$$O{x}_{i_1}\dots x_{i_k}:=\{\text{mathbbm}x=(x_1,\dots ,x_n):x_j=0,j\notin I\}$$

$$\{x_{i_1}=\dots =x_{i_k}=0\}:=\{\text{mathbbm}x=(x_1,\dots ,x_n):x_{i_1},\dots =x_{i_k}=0\}.$$

$$\text{upnu}(f_0)=n!V-\sum _{i=1}^n(n-1)!V_i+\sum _{\begin{array}{c}i,j=1\\ i<j\end{array}}^n(n-2)!V_{ij}-\dots +(-1{)}^n,$$

$${\text{uplambda}}^{\mathrm{nd}}(f_0):=\underset{(f_s)\in {\mathcal{D}}_0^{\mathrm{nd}}(f_0)}{\min }\text{uplambda}((f_s)),$$

$$\text{uplambda}(f_0)\le {\text{uplambda}}^{\mathrm{nd}}(f_0).$$

$${\lambda }^{\mathrm{nd}}(f_0)={\lambda }^{\mathrm{nd}}(\bar{f}).$$

$$\uplambda^{\operatorname{nd}}(f_{0})=\bigg{\{}\begin{array}[]{ll}\upmu(f_{0})-\upmu(f^{\operatorname{nd}}_{0}),&\text{if }\upmu(f_{0})-\upmu(f^{\operatorname{nd}}_{0})>0\\ \uplambda^{\operatorname{nd}}(f^{\operatorname{nd}}_{0}),&\text{if }\upmu(f_{0})-\upmu(f^{\operatorname{nd}}_{0})=0\end{array}.$$

$$\text{upnu}(f_0)-\text{upnu}(f_0^{\text{i}})=n!\mathrm{vol}(\Delta (\text{i},P_1,\dots ,P_n))>0,$$

$$(-1{)}^{n-(k+1)}(k+1)!\frac{\mathrm{vol}(W)p_n}{k+1}=(-1{)}^{n-1-k}k!\mathrm{vol}(W)p_n.$$

$$\text{upnu}(f_0)-\text{upnu}(f_0^{\text{i}})=(\text{upnu}{(}^n{f}_0)-\text{upnu}{(}^n{f}_0^{\text{i}}))(p_n-1).$$

$$f_s^{\text{i}}(\text{z}):=f_0(\text{z})+s{\text{z}}^{\text{i}}.$$

$${\text{uplambda}}^{\mathrm{nd}}(f_0)=\underset{\text{i}\in J}{\min }\text{uplambda}((f_s^{\text{i}})).$$

$$f_s(\text{z})=f_0(\text{z})+\sum _{\text{i}\in J}{a}_{\text{i}}(s)z^{\text{i}},$$

$$f_0(\text{z})=z_1^{p_1}+\dots +z_n^{p_n},p_i⩾2.$$

$$\lambda ((f_s^{\text{i}}))=\lambda ({(}^k{f}_s^{\text{i}}))(p_k-1).$$

$$f_0(\text{z})=z_1^{p_1}+\dots +z_n^{p_n},p_i⩾2,i=1,\dots ,n$$

$$\lambda ((f_s^{{\text{i}}_k}))={\lambda }_k(p_k-1).$$

$$\lambda ({(}^k{f}_s^{\text{i}}))⩾{\lambda }_k$$

$$\lambda ((f_s^{\text{i}}))=\lambda ({(}^k{f}_s^{\text{i}}))(p_k-1)⩾{\lambda }_k(p_k-1).$$

$$\underset{1⩽k⩽n}{\min }{\lambda }_k(p_k-1).$$

$$\lambda ((f_s^{\text{i}}))=n!\mathrm{vol}(\Delta (\text{i},P_1,\dots ,P_n))=p_1\dots p_n-i_1{p}_2\dots p_n-\dots -i_n{p}_1\dots p_{n-1}.$$

$$i_1{p}_2\dots p_n+\dots +i_n{p}_1\dots p_{n-1}=p_1\dots p_n-l$$

$$-i_1{p}_2\dots p_n-\dots -i_{n-1}{p}_1\dots p_{n-2}{p}_n+(p_n-i_n)p_1\dots p_{n-1}=l.$$

$$-i_1{p}_1^{\prime }-\dots -i_{n-1}{p}_{n-1}^{\prime }+\widetilde{i_n}{p}_n^{\prime }=l,l=1\dots ,{\lambda }^{\mathrm{hyp}}-1,$$

$$0<i_k<p_k,k=1,\dots ,n-1,0<\widetilde{i_n}<p_n.$$

$${\lambda }^{\mathrm{nd}}(f_0)=l_0.$$

$$j_1{p}_1^{\prime }+\dots +j_n{p}_n^{\prime }=1.$$

$$-i_1{p}_1^{\prime }-\dots -i_{n-1}{p}_{n-1}^{\prime }+i_n^{\prime }{p}_n^{\prime }=1$$

Full text

1. Introduction

Let $f_{0}:(\mathbbm{C}^{n},0)\rightarrow(\mathbbm{C},0)$ be an (isolated) singularity, i.e. $f_{0}$ is the germ at [math] of a holomorphic function having an isolated critical point at $0\in\mathbbm{C}^{n}$, and $0\in\mathbbm{C}$ as the corresponding critical value. More specifically, there exists a representative $\hat{f}_{0}:U\rightarrow\mathbbm{C}$ of $f_{0}$ holomorphic in an open neighborhood $U$ of the point $0\in\mathbbm{C}^{n}$ such that:

• $\bullet$

$\hat{f}_{0}(0)=0$,

• $\bullet$

$\nabla\hat{f}_{0}(0)=0$,

• $\bullet$

$\nabla\hat{f}_{0}(z)\neq 0$ for $z\in U\backslash\{0\}$,

where for a holomorphic function $f$ we put $\nabla f:=(\partial f/\partial z_{1},\ldots,\partial f/\partial z_{n})$.

The ring of germs of holomorphic functions of $n$ variables will be denoted by $\mathcal{O}_{n}$.

A deformation of the singularity $f_{0}$ is the germ of a holomorphic function $f=f(s,z):(\mathbbm{C}\times\mathbbm{C}^{n},0)\rightarrow(\mathbbm{C},0)$ such that:

• $\bullet$

$f(0,z)=f_{0}(z)$,

• $\bullet$

$f(s,0)=0$,

The deformation $f(s,z)$ of the singularity $f_{0}$ will also be treated as a family $(f_{s})$ of germs, putting $f_{s}(z):=f(s,z)$. Since $f_{0}$ is an isolated singularity, $f_{s}$ has also isolated singularities near the origin, for sufficiently small $s$ [8, Theorem 2.6 in Chap. I].

Remark 1**.**

Notice that in the deformation $(f_{s})$ there can occur in particular smooth germs, that is germs satisfying $\nabla f_{s}(0)\neq 0$. In this context, the symbol $\nabla f_{s}$ will always denote $\nabla_{z}f_{s}(z)$.

By the above assumptions, for every sufficiently small $s$, one can define a (finite) number ${\upmu}_{s}$ as the Milnor number of $f_{s}$, namely

[TABLE]

where $(\nabla f_{s})$ is the ideal in $\mathcal{O}_{n}$ generated by $\partial f_{s}/\partial z_{1},\ldots,\partial f_{s}/\partial z_{n}$.

Since the Milnor number is upper semi-continuous in the Zariski topology

[8, Prop. 2.57 in Chap. II], there exists an open neighborhood $S$ of the point $0\in\mathbbm{C}$ such that

• $\bullet$

${\upmu}_{s}=\operatorname{const.}$ for $s\in S\setminus\{0\}$,

• $\bullet$

${\upmu}_{0}\geqslant{\upmu}_{s}$ for $s\in S$.

The (constant) difference ${\upmu}_{0}-{\upmu}_{s}$ for $s\in S\setminus\{0\}$ will be called the jump of the deformation $(f_{s})$ and denoted by $\uplambda((f_{s}))$. The smallest non-zero value among the jumps of all deformations of the singularity $f_{0}$ will be called the jump (of the Milnor number) of the singularity $f_{0}$ and denoted by $\uplambda(f_{0})$.

The first general result concerning the jump was S. Guseĭn-Zade’s [9], who proved that there exist singularities $f_{0}$ for which $\uplambda(f_{0})>1$ and that for irreducible plane curve singularities it holds $\uplambda(f_{0})=1$. In [4] the authors proved that $\uplambda(f_{0})$ is not a topological invariant of $f_{0}$ but it is an invariant of the stable equivalence of singularities. The computation of $\uplambda(f_{0})$ is not an easy task. It is related to the problem of adjacency of classes of singularities. Only for few classes of singularities we know the exact value of $\uplambda(f_{0})$. For plane curve singularities ($n=2$) in the $X_{9}$ class (see [1] for terminology) that is singularities of the form $f^{a}_{0}(x,y):=x^{4}+y^{4}+ax^{2}y^{2}$, $a\in\mathbb{C}$, $a^{2}\not=4$, we have $\lambda(f^{a}_{0})=2$ (see [4]) and for specific homogeneous singularities $f^{d}_{0}(x,y)=x^{d}+y^{d}$, $d\geqslant 2$, we have $\lambda(f^{d}_{0})=\left[d/2\right]$, where $[x]$ is the integer part of $x$ (see [5]).

In the paper we consider a weaker problem: compute the jump $\uplambda^{\operatorname{nd}}(f_{0})$ of $f_{0}$ over all non-degenerate deformations of $f_{0}$ (i.e. the $f_{s}$ in deformations $(f_{s})$ of $f_{0}$ are non-degenerate singularities). Clearly, we always have $\uplambda(f_{0})\leqslant\uplambda^{\operatorname{nd}}(f_{0})$. Up to now, this problem has been studied only for plane curve singularities by A. Bodin [3], S. Brzostowski, T. Krasiński and J. Walewska [16], [17], [5]. We give a formula (more precisely an algorithm) in $n$-dimensional case for $\lambda^{\operatorname{nd}}(f_{0})$, where $f_{0}$ is non-degenerate, convenient and has its Newton diagram reduced to one $(n-1)$-dimensional face. Since for non-degenerate singularities $f_{s}$ of a deformation $(f_{s})$ of $f_{0}$ the Milnor number $\upmu(f_{s})$ depends only on the Newton diagram of $f_{s}$ and not on specific coefficients of $f_{s}$ (see Section 2) we may restrict considerations to the class of the Brieskorn-Pham singularities

[TABLE]

where $\textbf{z}=(z_{1},\ldots,z_{n})$.

Remark 2**.**

Let us note that we compute non-degenerate jump of a singularity in a fixed system of coordinates. The reason is that non-degeneracy depends on coordinate system. For instance, for singularity $f_{0}(x,y)=x^{3}+y^{3}$, $\lambda^{\operatorname{nd}}(f_{0})=2$ in the original system of coordinates while for $L:x=x^{\prime}-y^{\prime}$, $y=y^{\prime}$, we have $\lambda^{\operatorname{nd}}(f_{0}\circ L)=1$.

The problem of finding a minimal non-degenerate jump of a singularity in arbitrary system of coordinates is open.

2. Non-degenerate singularities

In this Section we recall the notion of non-degenerate singularities. Let $f_{0}(\textbf{z})=\sum_{\textbf{i}\in\mathbb{N}_{0}^{n}}a_{\textbf{i}}\textbf{z}^{\textbf{i}}$ be a singularity and $\operatorname{supp}(f_{0})=\{\textbf{i}\in\mathbb{N}_{0}^{n}:a_{\textbf{i}}\not=0\}$ the support of $f_{0}$. The Newton polyhedron $\Gamma_{+}(f_{0})\subset\mathbb{R}^{n}$ of $f_{0}$ is the convex hull of the set $\bigcup_{\textbf{i}\in\operatorname{supp}(f_{0})}(\textbf{i}+\mathbbm{R}_{+}^{n}).$ The finite set of compact faces (of all dimensions) of the boundary of $\Gamma_{+}(f_{0})$ constitutes the Newton diagram of $f_{0}$ and is denoted by $\Gamma(f_{0})$. The singularity $f_{0}$ is convenient if $\Gamma_{+}(f_{0})$ intersects all coordinate axes in $\mathbb{R}^{n}$. For each face $S\in\Gamma(f_{0})$ we define a weighted homogeneous polynomial $(f_{0})_{S}(\textbf{z})=\sum_{\textbf{i}\in S}a_{\textbf{i}}\textbf{z}^{\textbf{i}}.$ We call the singularity $f_{0}$ non-degenerate on $S\in\Gamma(f_{0})$ if the system of equations $\partial(f_{0})_{S}/\partial z_{i}(\textbf{z})=0$, $i=1,\ldots,n$ has no solutions in $(\mathbbm{C}^{*})^{n}$. The singularity $f_{0}$ is non-degenerate if $f_{0}$ is non-degenerate on every face $S\in\Gamma(f_{0})$.

In the sequel we put the following notations for coordinate hyperplanes in $\mathbb{R}^{n}$. Let $I=\{i_{1},\ldots,i_{k}\}\subset\{1,2,\ldots,n\}$. We denote

[TABLE]

and

[TABLE]

For a convenient singularity $f_{0}$ we define the Newton number $\upnu(f_{0})$ of $f_{0}$ by

[TABLE]

where $V$ is the $n$-dimensional volume under $\Gamma_{+}(f_{0})$, $V_{i}$ is the $(n-1)$-dimensional volume under $\Gamma_{+}(f_{0})$ on the hyperplane $\{x_{i}=0\}$, $V_{ij}$ is the $(n-2)$-dimensional volume under $\Gamma_{+}(f_{0})$ on the hyperplane $\{x_{i}=x_{j}=0\}$ and so on.

The importance of $\upnu(f_{0})$ has its source in the celebrated Kouchnirenko theorem.

Theorem 2.1** ([11]).**

If $f_{0}$ is a convenient singularity, then

1. (1)

${\upmu}(f_{0})\geqslant\upnu(f_{0})$, 2. (2)

if $f_{0}$ is non-degenerate then ${\upmu}(f_{0})=\upnu(f_{0})$.

Remark 3**.**

For non-convenient singularities the Kouchnirenko Theorem also holds provided $\nu(f_{0})$ is appropriately defined (see for instance [6]). It is interesting that both conditions of item 2. of the Theorem are equivalent in $2$-dimensional case by A. Płoski [15]. For weaker non-degeneracy conditions equivalent to the equality $\mu(f_{0})=\nu(f_{0})$ in $n$-dimensional case see recent preprint by P. Mondal [12].

3. Non-degenerate jump of Milnor numbers of singularities

Let $f_{0}\in\mathcal{O}_{n}$ be a singularity. A deformation $(f_{s})$ of $f_{0}$ is called non-degenerate if $f_{s}$ is non-degenerate for $s\neq 0$. The set of all non-degenerate deformations of the singularity $f_{0}$ will be denoted by $\mathcal{D}^{\operatorname{nd}}(f_{0})$. Non-degenerate jump $\uplambda^{\operatorname{nd}}(f_{0})$ of the singularity $f_{0}$ is the minimal of non-zero jumps over all non-degenerate deformations of $f_{0}$, which means

[TABLE]

where by $\mathcal{D}^{\operatorname{nd}}_{0}(f_{0})$ we denote all non-degenerate deformations $(f_{s})$ of $f_{0}$ for which $\uplambda((f_{s}))\neq 0$.

Obviously

Proposition 3.1**.**

For each singularity $f_{0}$ we have the inequality

[TABLE]

From the Kouchnirenko Theorem we get

Corollary 3.2**.**

*If $f_{0}$ and $\tilde{f_{0}}$ are non-degenerate and convenient singularities and

$\Gamma(f_{0})=\Gamma(\tilde{f_{0}})$ then $\uplambda^{\operatorname{nd}}(f_{0})=\uplambda^{\operatorname{nd}}(\tilde{f_{0}})$.*

Corollary 3.3**.**

If $f_{0}$ is a convenient and non-degenerate singularity such that the Newton diagram $\Gamma(f_{0})$ has only one $(n-1)$-dimensional face which intersects the axes $Ox_{1},\ldots,Ox_{n}$ in points $(p_{1},0,\ldots,0),\ldots,(0,\ldots,0,p_{n})$ respectively, then for the Brieskorn-Pham singularity $\bar{f}(\textbf{z})=z_{1}^{p_{1}}+\ldots+z_{n}^{p_{n}}$ we have

[TABLE]

In investigations concerning $\uplambda^{\operatorname{nd}}(f_{0})$ we may restrict our attention to non-dege- nerate $f_{0}$ because the non-degenerate jump for degenerate singularities can be found using the proposition below (cf. [3, Lemma 5]). For a given $f_{0}$ let $f^{\operatorname{nd}}_{0}$ denote any non-degenerate singularity for which $\Gamma(f_{0})=\Gamma(f^{\operatorname{nd}}_{0})$. Such singularities always exist.

Proposition 3.4**.**

If $f_{0}$ is degenerate then

[TABLE]

**Proof. ** This follows from the fact that a generic small perturbation of coefficients of these monomials of $f_{0}$ which correspond to points belonging to $\bigcup\Gamma(f_{0})$ (which are finite in number) give us non-degenerate singularities with the same Newton polyhedron as $f_{0}$. $\Box$

Remark 4**.**

By the Płoski theorem ([14, Lemma 2.2], [15, Theorem 1.1]), for degenerate plane curve singularities ($n=2$) the second possibility in Proposition 3.4 is excluded.

A crucial role in the search for the formula for $\uplambda^{\operatorname{nd}}(f_{0})$ will be played by the monotonicity of the Newton number with respect to the Newton polyhedron. Namely, J. Gwoździewicz [10], M. Furuya [7] and C. Biviá-Ausina [2] proved.

Theorem 3.5** (Monotonicity Theorem).**

Let $f_{0},\tilde{f_{0}}\in\mathcal{O}_{n}$ be two convenient singularities such that $\Gamma_{+}(f_{0})\subset\Gamma_{+}(\tilde{f_{0}})$. Then $\upnu(f_{0})\geqslant\upnu(\tilde{f_{0}})$.

We need a stronger version of the above theorem in the particular case $f_{0}$ has only one $(n-1)$-dimensional face and $\tilde{f_{0}}$ differs from $f_{0}$ in one monomial.

Theorem 3.6**.**

If $f_{0}$ is a convenient and non-degenerate singularity such that the Newton diagram $\Gamma(f_{0})$ has only one $(n-1)$-dimensional face and $f^{\textup{{i}}}_{0}(\textbf{z}):=f_{0}(\textbf{z})+\textbf{z}^{\textup{{{i}}}}$, where the lattice point $\textup{{i}}\not=0$ lies under $\Gamma_{+}(f_{0})$ i.e. $0\not=\textup{{i}}\in\mathbb{N}_{0}^{n}\setminus\Gamma_{+}(f_{0})$, then $\upnu(f_{0})>\upnu(f^{\textup{{i}}}_{0})$.

**Proof. ** By the Kouchnirenko Theorem we may consider only the Brieskorn-Pham singularities (1). We will use induction with respect to the number $n$ of variables. For $n=1$ we have $f_{0}(\textbf{z})=z^{p_{1}}$, $p_{1}\geqslant 2$, and $f^{\textbf{i}}_{0}(\textbf{z})=z^{p_{1}}+z^{i}$, $1\leqslant i<p_{1}$. Hence $\upnu(f_{0})=p_{1}-1$ and $\upnu(f^{\textbf{i}}_{0})=i-1$. This gives $\upnu(f_{0})>\upnu(f^{\textbf{i}}_{0})$.

Suppose the theorem is true for any Brieskorn-Pham singularity in $(n-1)$ variables. Let now $f_{0}(\textbf{z})=z_{1}^{p_{1}}+\ldots+z_{n}^{p_{n}}$, $p_{i}\geqslant 2$, $i=1,\ldots,n$. Let $P_{i}=(0,\ldots,p_{i},\ldots,0)$.

If i does not lie on any coordinate hyperplane $\{x_{i}=0\}$ then $\Gamma_{+}(f_{0})$ and $\Gamma_{+}(f^{\textbf{i}}_{0})$ are identical on all these hyperplanes (see Figure 1(a)) and hence

[TABLE]

where $\Delta(\textbf{i},P_{1},\ldots,P_{n})$ is the $n$-dimensional simplex with vertices $\textbf{i},P_{1},\ldots,P_{n}$.

Take now a lattice point $\textbf{i}\not=0$ under $\Gamma_{+}(f_{0})$ which lies on a coordinate hyperplane. For simplicity we may assume $\textbf{i}\in Ox_{1}\ldots x_{n-1}$ i.e. $\textbf{i}=(i_{1},\ldots,i_{n-1},0)$. Two possible cases in dimension three are illustrated in Figure 1(b),(c). By definition of the Newton number the difference $\upnu(f_{0})-\upnu(f^{\textbf{i}}_{0})$ is equal to the sum of volumes of differences of $\Gamma_{+}(f_{0})$ and $\Gamma_{+}(f^{\textbf{i}}_{0})$ on all hyperplanes $Ox_{i_{1}}\ldots x_{i_{l}}$ multiplied by $(-1)^{n-l}l!$ $({i_{1}},\ldots,{i_{l}}\text{ are different and }l=1,\ldots,n)$. First we consider these differencies in hyperplanes which do not contain the axis $Ox_{n}$. They are equal to the sum of volumes of differencies for truncated singularities ${}^{n}f_{0}(\textbf{z}):=z_{1}^{p_{1}}+\ldots+z_{n-1}^{p_{n-1}}$ and ${}^{n}f^{\textbf{i}}_{0}(\textbf{z}):={{}^{n}f_{0}(\textbf{z})}+\textbf{z}^{\textbf{i}}$ but taken with opposite signs. So, the sum of this differencies is equal to $-(\upnu(^{n}f_{0})-\upnu(^{n}f^{\textbf{i}}_{0}))$. Now we consider differencies in coordinate hyperplanes which contain the axis $Ox_{n}$. Each such difference in a coordinate hyperplane $Ox_{i_{1}}\ldots x_{i_{k}}x_{n}$ is a pyramid whose base is equal to the difference, say $W$, of $\Gamma_{+}(f_{0})$ and $\Gamma_{+}(f^{\textbf{i}}_{0})$ in $Ox_{i_{1}}\ldots x_{i_{k}}$ and the vertex is equal to $P_{n}=(0,\ldots,0,p_{n})$. Hence, in $\upnu(f_{0})-\upnu(f^{\textbf{i}}_{0})$ this pyramid will give the term

[TABLE]

But the latter expression is precisely the term in $(\upnu(^{n}f_{0})-\upnu(^{n}f^{\textbf{i}}_{0}))p_{n}$ associated to the difference $W$. Together we get

[TABLE]

Since $p_{n}>1$ and by induction hypothesis $\upnu(^{n}f_{0})-\upnu(^{n}f^{\textbf{i}}_{0})$ is positive, we get the assertion of the theorem. $\Box$

Now we define specific non-degenerate deformations of a convenient and non-degenerate singularity $f_{0}\in\mathcal{O}_{n}$ (in particular for Brieskorn-Pham singularities). Denote by $J$ the set of lattice points (points with integer coordinates) $\textbf{i}\not=0$ lying under the Newton diagram of $f_{0}$ i.e. $0\not=\textbf{i}\in\mathbb{N}_{0}^{n}\setminus\Gamma_{+}(f_{0})$. Obviously, $J$ is a finite set. For $\textbf{i}=(i_{1},\ldots,i_{n})\in J$ we define the monomial deformation $(f_{s}^{\textbf{i}})_{s\in\mathbb{C}}$ of $f_{0}$ by the formula

[TABLE]

Proposition 3.7**.**

For every $\textup{{i}}\in J$ the deformation $(f^{\textup{{i}}}_{s})$ of $f_{0}$ is convenient and non-degenerate for all sufficiently small $|s|$.

**Proof. ** See [11] or [13, Appendix].

$\Box$

Combining the Monotonicity Theorem with the above proposition we reach the conclusion that in order to find $\uplambda^{\operatorname{nd}}(f_{0})$ it is enough to consider only the non-degenerate deformations of the type $(f^{\textup{{i}}}_{s})$.

Theorem 3.8**.**

If $f_{0}$ is a convenient and non-degenerate singularity such that the Newton diagram $\Gamma(f_{0})$ has only one $(n-1)$-dimensional face, then

[TABLE]

**Proof. ** By the Kouchnirenko theorem it suffices to consider non-degenerate deformations of $f_{0}$ of the form

[TABLE]

where $a_{\textup{{i}}}(s)$ are holomorphic at $0\in\mathbbm{C}$ and $a_{\textup{{i}}}(0)=0$. Then by the Monotonicity Theorem and Theorem 3.6 we may restrict the scope of deformations (2) to deformations with only one term added i.e. to the deformations $(f^{\textup{{i}}}_{s})$ for $\textup{{i}}\in J$. $\Box$

4. An algorithm for the non-degenerate jump of Brieskorn-Pham singularities

In this Section we give an algorithm for calculating $\uplambda^{\operatorname{nd}}(f_{0})$ for a non-degenerate and convenient singularity with one $(n-1)$-dimensional face of its Newton diagram. By Corollary 3.3 we may assume that $f_{0}$ is a Brieskorn-Pham singularity

[TABLE]

We have $\upmu(f_{0})=\prod\limits_{i=1}^{n}(p_{i}-1)$. By Theorem 3.8 to find $\lambda^{\operatorname{nd}}(f_{0})$ we have to compare the jumps of all monomial deformations $(f^{\textbf{i}}_{s})_{s\in\mathbb{C}}$ where $\textbf{i}\not=0$ is a lattice point lying under $\Gamma_{+}(f_{0})$. First we prove a lemma relating the jumps of monomial deformations of the Brieskorn-Pham singularities to the jumps of their "truncations".

Lemma 4.1**.**

Let $f_{0}(\textbf{z})=z_{1}^{p_{1}}+\ldots+z_{n}^{p_{n}}$ be a Brieskorn-Pham singularity. Define ${}^{k}f_{0}(\textbf{z}):=z_{1}^{p_{1}}+\ldots+z_{k-1}^{p_{k-1}}+z_{k+1}^{p_{k+1}}+\ldots+z_{n}^{p_{n}}$, $k=1,\ldots,n$. Let $\textup{{i}}\not=0$ be a lattice point under $\Gamma_{+}(f_{0})$ lying in the hyperplane $\{x_{k}=0\}$. Then

[TABLE]

**Proof. ** The proof follows the line of the proof of Theorem 3.6. For simplicity of notations we take $k=n$, i.e. $\textbf{i}=(i_{1},\ldots,i_{n-1},0)$. Since $\lambda((f_{s}^{\textbf{i}}))=\upnu(f_{0})-\upnu(f_{s}^{\textbf{i}})$, $s\not=0$, by definition of the Newton number the difference $\upnu(f_{0})-\upnu(f^{\textbf{i}}_{s})$ is equal to the sum of volumes of differences of $\Gamma_{+}(f_{0})$ and $\Gamma_{+}(f^{\textbf{i}}_{s})$ on all hyperplanes $Ox_{i_{1}}\ldots x_{i_{l}}$ multiplied by $(-1)^{n-l}l!$ $({i_{1}},\ldots,{i_{l}}\text{ are different and }l=1,\ldots,n)$. Similarly as in the proof of Theorem 3.6 we divide all differencies in hyperplanes into two kinds: in those hyperplanes which do not contain the axis $Ox_{n}$ and those which do contain the axis $Ox_{n}$. The sum of the first ones is equal to the number $-\lambda((^{n}f_{s}^{\textbf{i}}))$ while the second ones $\lambda((^{n}f_{s}^{\textbf{i}}))p_{n}$. Together we get equality (4) for $k=n$. $\Box$

Now we present an inductive algorithm for $\lambda^{\operatorname{nd}}(f_{0})$ where $f_{0}$ is any Brieskorn-Pham singularity

[TABLE]

with respect to the number of variables $n$. In each step we give the formula for $\lambda^{\operatorname{nd}}(f_{0})$ and simultaneously the point $\textbf{i}\in J$ (precisely the deformation $(f^{\textbf{i}}_{s})$) realizing this jump. For $n=1$ we have $f_{0}(z_{1})=z_{1}^{p_{1}}$, $p_{1}\geqslant 2$ and $J=\{1,2,\ldots,p_{1}-1\}$. We have $\lambda^{\operatorname{nd}}(f_{0})=1$ and the point $\textbf{i}=(p_{1}-1)$ realizes this jump.

Assume that for all Brieskorn-Pham singularities $f_{0}$ in $(n-1)$-variables we can compute $\lambda^{\operatorname{nd}}(f_{0})$ and can indicate the point i realizing this jump.

Take the Brieskorn-Pham singularity $f_{0}(\textbf{z})=z_{1}^{p_{1}}+\ldots+z_{n}^{p_{n}},p_{i}\geqslant 2,n\geqslant 2$, in $n$ variables. By Theorem 3.8 we have to compare the jumps of all monomial deformations $(f_{s}^{\textbf{i}})$, where $\textbf{i}\not=0$ is a lattice point lying under $\Gamma_{+}(f_{0})$.

A. First we consider points $\textbf{i}\not=0$ lying in coordinate hyperplanes of dimension $(n-1)$. By induction hypotheses on each coordinate hyperplane

$Ox_{1}\ldots x_{k-1}x_{k+1}\ldots x_{n}$ there exists a point $\textbf{i}_{k}$ realizing non-degenerate jump of the truncated singularity ${}^{k}f_{0}=z_{1}^{p_{1}}+\ldots+z_{k-1}^{p_{k-1}}+z_{k+1}^{p_{k+1}}+\ldots+z_{n}^{p_{n}}$ in $(n-1)$ variables. Denote this jump by $\lambda_{k}:=\lambda^{\operatorname{nd}}(^{k}f_{0})=\lambda((^{k}f_{s}^{\textbf{i}_{k}}))$. By the Lemma 4.1 the jump of the deformation $(f_{s}^{\textbf{i}_{k}})$ is given by the formula

[TABLE]

This formula implies that there is no "better" point in the hyperplane

$Ox_{1}\ldots x_{k-1}x_{k+1}\ldots x_{n}$ which could give a monomial deformation of $f_{0}$ with smaller jump. In fact, for any point $\textbf{i}\in J\cap Ox_{1}\ldots x_{k-1}x_{k+1}\ldots x_{n}$ we would have

[TABLE]

and hence again by the Lemma 4.1

[TABLE]

Hence doing the same for all hyperplanes $Ox_{1}\ldots x_{k-1}x_{k+1}\ldots x_{n}$, $k=1,\ldots,n$, we get that the minimum of the jumps of monomial deformations associated to points lying in these hyperplanes is equal to

[TABLE]

Denote this number by $\lambda^{\operatorname{hyp}}$.

B. Now we consider monomial deformations of $f_{0}$ associated to points $\textbf{i}\not=0$ under $\Gamma_{+}(f_{0})$ not lying in coordinate hyperplanes i.e. lying in the interior of the simplex $\Delta(P_{0},P_{1},\ldots,P_{n})$, where $P_{0}=(0,\ldots,0)$, $P_{1}=(p_{1},0,\ldots,0)$,$\ldots$, $P_{n}=(0,\ldots,0,p_{n})$. Such a point $\textbf{i}=(i_{1},\ldots,i_{n})$ satisfies the conditions:

$0<i_{k}<p_{k}$, $k=1,\ldots n$, 2. 2.

$\dfrac{i_{1}}{p_{1}}+\ldots+\dfrac{i_{n}}{p_{n}}<1$ (or equivalently $i_{1}p_{2}\ldots p_{n}+\ldots+i_{n}p_{1}\ldots p_{n-1}<p_{1}\ldots p_{n}$).

For such a point i we obviously have

[TABLE]

This follows from the fact that $\operatorname{vol}(\Delta(P_{0},\ldots,P_{n}))=\dfrac{p_{1}\ldots p_{n}}{n!}$ and

$\operatorname{vol}(\Delta(P_{0},\ldots,P_{k-1},\textbf{i},P_{k+1},\ldots,P_{n}))=\dfrac{i_{k}p_{1}\ldots p_{k-1}p_{k+1}\ldots p_{n}}{n!}$, $k=1,\ldots,n$. So the problem is to find minimal $l$ in the set $\{1,\ldots,\lambda^{\operatorname{hyp}}-1\}$ for which there exist solutions $i_{1},\ldots,i_{n}$ of the diophantine equation

[TABLE]

satisfying conditions 1. and 2. First, we transform (5) into the more useful form

[TABLE]

If we denote $p^{\prime}_{k}:=p_{1}\ldots p_{k-1}p_{k+1}\ldots p_{n}$, $k=1,\ldots,n$, then we reduce the problem to find integer solutions $i_{1},\ldots,i_{n-1},\widetilde{i_{n}}$ of the equations

[TABLE]

satisfying

[TABLE]

Then for such a minimal $l_{0}\in\{1,\ldots,\lambda^{\operatorname{hyp}}-1\}$ for which the solution of (7) with conditions exists we have

[TABLE]

We give a simple algorithm (using only Euclid’s algorithm) to solve the equation (7) with conditions (8) only in the case $\operatorname{GCD}(p_{i},p_{j})=1$, $i,j=1,\ldots,n$, $i\not=j$. In general case we also may compute $\lambda^{\operatorname{nd}}(f_{0})$ but modulo finding solutions of (7) satisfying (8).

Assume $f_{0}=z_{1}^{p_{1}}+\ldots+z_{n}^{p_{n}}$, $p_{i}\geqslant 2$, $\operatorname{GCD}(p_{i},p_{j})=1$, for $i\not=j$. First we consider the equation (7) for $l=1$. Since $p_{1},\ldots,p_{n}$ are pairwise prime therefore $\operatorname{GCD}(p^{\prime}_{1},\ldots,p^{\prime}_{n})=1$. Hence by Euclid’s algorithm we find integers $j_{1},\ldots,j_{n}\in\mathbb{Z}$ such that

[TABLE]

By simple transformations we get another identity

[TABLE]

where $0<i_{k}<p_{k}$, $k=1,\ldots,n-1$, $i^{\prime}_{n}\in\mathbb{N}$. If $i^{\prime}_{n}<p_{n}$ then for $\textbf{i}=(i_{1},\ldots,i_{n-1},p_{n}-i^{\prime}_{n})$ we have

[TABLE]

and consequently

[TABLE]

If $i^{\prime}_{n}\geqslant p_{n}$ then we claim there is no solution to our problem for $l=1$. Assume to the contrary that $j_{1},\ldots,j_{n}$ is such a solution i.e.

[TABLE]

Hence and by (9) we obtain

[TABLE]

Since $p_{1}|p^{\prime}_{k}$, $k=2,\ldots,n$, then $p_{1}|(j_{1}-i_{1})p^{\prime}_{1}$. But $p_{1}$ is relatively prime to $p_{2},\ldots,p_{n}$. Hence $p_{1}|(j_{1}-i_{1})$. But $|j_{1}-i_{1}|<p_{1}$ which implies $j_{1}=i_{1}$.

Similarly $j_{2}=i_{2},\ldots,j_{n-1}=i_{n-1}$. This implies $i^{\prime}_{n}=p_{n}-j_{n}<p_{n}$, which contradicts our supposition $i^{\prime}_{n}\geqslant p_{n}$.

So in this case $\lambda^{\operatorname{nd}}(f_{0})>1$ and we repeat our considerations for $l=2$ etc. up to $\lambda^{\operatorname{hyp}}-1$. If in one of this steps we find solutions $i_{1},\ldots,i_{n}$ of (7) satisfying conditions (8) we stop the procedure and in this case $\lambda^{\operatorname{nd}}(f_{0})=l$ and the monomial deformation $(f^{\textbf{i}}_{s})$ for $\textbf{i}=(i_{1},\ldots,i_{n-1},p_{n}-i_{n})$ realizes the jump. If the above search fails we conclude

[TABLE]

We may sum up the above considerations in the following theorem.

Theorem 4.2**.**

*Let $f_{0}\in\mathcal{O}^{n}$ be a convenient and non-degenerate singularity with only one

$(n-1)$-dimensional face of its Newton diagram. Assume the vertices (on axes) of this face are $(p_{1},0,\ldots,0),\ldots,(0,\ldots,p_{n}),\;p_{i}\geqslant 2$. Then the non-degenerate jump $\uplambda^{\operatorname{nd}}(f_{0})$ of $f_{0}$ can be inductively (with respect to $n$) calculated as follows. For $n=1$, $\uplambda^{\operatorname{nd}}(f_{0})=1$. For $n\geqslant 2$ let $\uplambda_{k}$ be the non-degenerate jump of the restriction ${}^{k}f_{0}$ of $f_{0}$ to the hyperplane $\{z_{k}=0\}$ and $\uplambda^{\operatorname{hyp}}:=\min\limits_{1\leqslant k\leqslant n}(\uplambda_{k}(p_{k}-1))$. Then putting $p^{\prime}_{k}:=p_{1}\ldots p_{k-1}p_{k+1}\ldots p_{n}$, $k=1,\ldots,n$ we have*

[TABLE]

In the first case the jump is realized by the monomial deformation $(f^{\textup{{i}}}_{s})$ where $\textup{{i}}=(-j_{1},\ldots,-j_{n-1},p_{n}-j_{n})$ and in the second case by $(f^{\textup{{i}}}_{s})$ where i is the point in a coordinate hyperplane $\{x_{k_{0}}=0\}$ for which $\uplambda^{\operatorname{hyp}}=\uplambda_{k_{0}}(p_{k_{0}}-1)$, realizing the jump for the restriction ${}^{k_{0}}f_{0}$ of $f_{0}$ to the hyperplane $\{z_{k_{0}}=0\}$.

Moreover in the case $\operatorname{GCD}(p_{i},p_{j})=1$ for $i\not=j$ the jump $\uplambda^{\operatorname{nd}}(f_{0})$ can be computed using only Euclid’s algorithm.

Example 1** (Bodin [3], Walewska [16]).**

For $n=2$ and $f_{0}(\textbf{z})=z_{1}^{p_{1}}+z_{2}^{p_{2}}$ putting $d:=\operatorname{GCD}(p_{1},p_{2})$ we have from the above theorem

[TABLE]

In fact, we have $\uplambda_{1}=\uplambda_{2}=1$, $\uplambda^{\operatorname{hyp}}=\min(p_{1}-1,p_{2}-1)$ and we always have the solution $j_{1},j_{2}$ of $j_{1}p_{2}+j_{2}p_{1}=d$, $0<-j_{1}<p_{1}$, $0<j_{2}<p_{2}$.

Example 2**.**

Let $f_{0}(\textbf{z})=z_{1}^{11}+z_{2}^{6}+z_{3}^{5}$. We have $p_{1}=11$, $p_{2}=6$, $p_{3}=5$ and by Example 1 $\uplambda_{1}=\uplambda_{2}=\uplambda_{3}=1$, $\uplambda^{\operatorname{hyp}}=4$. By the algorithm given in the proof we easily find solutions of diophantine equations in the formula (10) for $l=1,2,3$

[TABLE]

[TABLE]

[TABLE]

Only the last equation fulfills the condition in the formula. Hence

[TABLE]

and the jump is realized by the monomial deformation

[TABLE]

Acknowledgement

The first author was partially supported by the Polish National Science Centre (NCN), Grant No. 2012/07/B/ST1/03293.