2. Preliminaries on Newton polyhedra

In this section we study geometric properties of pairs of Newton polyhedra having the same Newton number, one contained in the other. In this article we study the deformations of hypersurfaces $ {\mathbb {C}}^{n+1}_{o}$, whereby the natural things would be to study polytopes in $ {\mathbb {R}}^{n+1}$, $n\geq 1$. Nevertheless, in order to avoid complicating the notations unnecessarily, we work with polytopes in $ {\mathbb {R}}^{n}$, $n\geq 2$.

Given an affine subspace $H$ of $ {\mathbb {R}}^n$, a convex polytope in $H$ is a non-empty set $P$ given by the intersection of $H$ with a finite set of half spaces of $ {\mathbb {R}}^n$. In particular, a compact convex polytope can be seen as the convex hull of a finite set of points in $ {\mathbb {R}}^n$. The dimension of a convex polytope is the dimension of the smallest affine subspace of $ {\mathbb {R}}^n$ that contains it. We say that $P$ is a polyhedron (respectively, compact polyhedron) if $P$ can be decomposed into a finite union of convex (respectively, compact convex) polytopes of disjoint interiors. We will say that $P$ is of pure dimension $n$ if $P$ is a finite union of $n$-dimensional convex polytopes. A hyperplane $K$ of $ {\mathbb {R}}^n$ is supporting $P$ if one of the two closed half spaces defined by $K$ contains $P$. A subset $F$ of $P$ is called a face of $P$ if it is either $\emptyset$, $P$ itself, or the intersection of $P$ with a supporting hyperplane. A face $F$ of $P$ of dimension $0\leq d \leq \dim (P)-1$ is called $d$-dimensional face. In the case that $d$ is $0$ or $1$, $F$ is called a vertex or edge, respectively.

An $n$-dimensional simplex $\Delta$ is a compact convex polytope generated by $n+1$ points of $ {\mathbb {R}}^n$ in general position.

Given an $n$-dimensional compact polyhedron $P\subset {\mathbb {R}}^{n}_{\geq 0}$ , the Newton number of $P$ is defined by

\[ \nu(P):=n!V_n(P)-(n-1)!V_{n-1}(P)\cdots (-1)^{n-1}V_1(P)+(-1)^nV_0(P), \]

where $V_n(P)$ is the volume of $P$, $V_{k}(P)$, $1\leq k\leq n-1$, is the sum of the $k$-dimensional volumes of the intersection of $P$ with the coordinate planes of dimension $k$, and $V_0(P)=1$ (respectively, $V_0(P)=0$) if $o\in P$ (respectively, $o\notin P$), where $o$ is the origin of $ {\mathbb {R}}^n$. In this section, we are interested in studying the monotonicity of the Newton Number, we always consider the case when $P$ is compact.

Let $I\subset \{1,2,\ldots,n\}$. We define the following sets:

\[ {\mathbb{R}}^{I}:=\{(x_1,\ldots,x_n)\in {\mathbb{R}}^{n}:x_i=0\ \text{if}\ i\notin I\},\quad {\mathbb{R}}_{I}=\{(x_1,\ldots,x_n)\in {\mathbb{R}}^{n}:x_i=0\ \text{if}\ i\in I\}. \]

Given a polyhedron $P$ in $ {\mathbb {R}}^n$, we write $P^I:=P\cap {\mathbb {R}}^{I}$. Consider an $n$-dimensional simplex $\Delta \subset {\mathbb {R}}^n_{\geq 0}$. A full supporting coordinate subspace of $\Delta$ is a coordinate subspace $ {\mathbb {R}}^{I}\subset {\mathbb {R}}^n$ such that $\dim \Delta ^I=|I|$. In [Reference FuruyaFur04], Furuya proved that there exists a unique full-supporting coordinate subspace of $\Delta$ of minimal dimension. We call this subspace the minimal full-supporting coordinate subspace (m.f.-s.c.s.) of $\Delta$.

We denote by $\operatorname {Ver}(P)$ the set of vertices of $P$.

The next result gives us a way of calculating the Newton number of certain polyhedra using projections.

Proposition 2.1 [Reference FuruyaFur04]

Let $o\notin P\subset {\mathbb {R}}^{n}_{\geq 0}$ be a compact polyhedron that is a finite union of $n$-simplices $\Delta _i$, $1\leq i\leq m$, that satisfy

\[ \operatorname{Ver}(\Delta_i)\subset \operatorname{Ver}(P). \]

Assume that there exists $I\subset \{1,2,\ldots,n\}$ such that $ {\mathbb {R}}^I$ is the m.f.-s.c.s. of $\Delta _i$ and $P^I=\Delta _{i}^{I}$ for all $1\leq i\leq m$. Then $\nu (P)= |I|!V_{|I|}(P^I)\nu (\pi _I(P))$ where $\pi _I: {\mathbb {R}}^n\rightarrow {\mathbb {R}}_{I}$ is the projection map.

Let $\mathcal {E}:=\{e_1,e_2,\ldots,e_n\}\subset {\mathbb {Z}}_{\geq 0}^n$ be the standard basis of $ {\mathbb {R}}^n$. Let

\[ P\subset {\mathbb{R}}_{\geq 0}^n \]

be a polyhedron of pure dimension $n$. Consider the following conditions:

1. (i) $o\in P$;

2. (ii) $P^J$ is homeomorphic to a $|J|$-dimensional closed disk for each $J\subset \{1,\ldots,n\}$;

3. (iii) Let $I\subset \{1,\ldots,n\}$ be a non-empty subset; if $(\alpha _1,\ldots,\alpha _n)\in \operatorname {Ver}(P)$, then for each $i\in I$ we must have either $\alpha _i\geq 1$ or $\alpha _i=0$ (recall that the $\alpha _i$ are real numbers that need not be integers).

We say that $P$ is pre-convenient (respectively, $I$-convenient) if it satisfies conditions (i) and (ii) (respectively, conditions (i), (ii), and (iii)). In the case when $I:=\{1,\ldots,n\}$ we simply say that $P$ is convenient instead of $I$-convenient.

Given a closed discrete set $S\subset {\mathbb {R}}^{n}_{\geq 0} {\setminus} \{o\}$, denote by $\Gamma _{+}(S)$ the convex hull of the set $\bigcup _{\alpha \in S}(\alpha + {\mathbb {R}}_{\geq 0}^n)$. The polyhedron $\Gamma _{+}(S)$ is called the Newton polyhedron associated to $S$. The Newton boundary of $\Gamma _{+}(S)$, denoted by $\Gamma (S)$, is the union of the compact faces of $\Gamma _{+}(S)$. Let $\operatorname {Ver}(S):=\operatorname {Ver}(\Gamma (S))$ denote the set of vertices of $\Gamma (S)$

Remark 2.2 Note that when we refer to vertices of $\Gamma (S)$ we are speaking about zero-dimensional faces of $\Gamma (S)$. For example, if

\[ S:=\{(0,3), (1,2), (2,1), (4,0), (3,1)\}, \]

then $\operatorname {Ver}(S)=\{(0,3),(2,1), (4,0)\}$.

We say that a closed discrete set $S\subset {\mathbb {R}}^{n}_{\geq 0} {\setminus} \{o\}$ is pre-convenient (respectively, $I$-convenient) if $\Gamma _{-}(S):=\overline { {\mathbb {R}}_{\geq 0}^n{\setminus} \Gamma _{+}(S)}$ is pre-convenient (respectively, $I$-convenient). The Newton number of a pre-convenient closed discrete set $S\subset {\mathbb {R}}^{n}_{\geq 0} {\setminus} \{o\}$ is

\[ \nu(S):=\nu(\Gamma_{-}(S)). \]

Note that this number can be negative. In the case when $P$ is the polyhedron $\Gamma _-(S)$ associated with a closed discrete set $S$, condition (i) holds automatically and condition (ii) can be replaced by the following:

1. (ii′) for each $e\in \mathcal {E}$ there exists $m>0$ such that $me\in \operatorname {Ver}(S)$.

Consider a convergent power series $g\in {\mathbb {C}}\{x_1,\ldots,x_n\}$:

\[ g(x)=\sum_{\alpha\in Z}\ a_{\alpha}x^{\alpha},\quad Z:={\mathbb{Z}}_{\geq 0}^n{\setminus} \{o\}. \]

We define $\Gamma _{+}(g)=\Gamma _+(\operatorname {Supp}(g))$ and $\Gamma (g)=\Gamma (\operatorname {Supp}(g))$. We say that $g$ is a convenient power series if for all $e\in \mathcal {E}$ there exists $m>0$ such that $me\in \operatorname {Supp}(g)$.

Observe that the closed discrete set $\operatorname {Supp}(g)$ is convenient if and only if the power series $g$ is convenient. We use the following notation: $\operatorname {Ver}(g):=\operatorname {Ver}(\operatorname {Supp}(g))$, and $\nu (g)=\nu (\operatorname {Supp}(g))$.

For a set $I\subset \{1,\ldots,n\}$, we write $I^c:=\{1,\ldots,n\}{\setminus} I$. The following result gives us a criterion for the positivity of the Newton number of certain polyhedra.

Proposition 2.6 Let $o\notin P$ be a pure $n$-dimensional compact polyhedron such that there exists $I\subset \{1,\ldots,n\}$ such that $\dim (P^J)<|J|$ (respectively, $P^J$ is homeomorphic to a $|J|$-dimensional closed disk) for all $I\not \subset J$ (respectively, $I\subset J$). Assume that if

\[ (\beta_1,\ldots,\beta_n)\in \operatorname{Ver}(P) \]

then for each $i\in I^c$ we have $\beta _i\geq 1$ or $\beta _i=0$. Then there exists a sequence of sets $I\subset I_1,I_2,\ldots,I_m\subset \{1,\ldots,n\}$, and of polyhedra $Z_{i}$, $1\leq i \leq m$, such that:

1. (i) $P=\bigcup _{i=1}^{m}Z_i$;

2. (ii) $\nu (P)=\sum _{i=1}^{m}\nu (Z_i)$;

3. (iii) $\nu (Z_{i})= |I_i|!V_{|I_i|}(Z_{i}^{I_{i}})\nu (\pi _{I_i}(Z_{i}))\geq 0$.

In particular, $\nu (P)\geq 0$.

Given $S\subset {\mathbb {R}}_{\geq }^{n}{\setminus} \{o\}$ and $R\subset {\mathbb {R}}^n_{\geq 0}$, we denote $S(R):=S\cup R$.

Proof. The method of proof that we use is similar to the proof of Theorem $2.3$ of [Reference FuruyaFur04].

As $P$ is a pure $n$-dimensional compact polyhedron, there exists a finite simplicial subdivision $\Sigma$ of $P$ such that:

1. (i) if $\Delta \in \Sigma$, then $\dim \Delta =n$;

2. (ii) for all $\Delta \in \Sigma$, $\operatorname {Ver}(\Delta )\subset \operatorname {Ver}(P)$;

3. (iii) given $\Delta, \Delta '\in \Sigma$, we have $\dim (\Delta \cap \Delta ')< n$ whenever $\Delta \neq \Delta '$.

Let $\mathcal {S}$ be the set formed by all the subsets $I'\subset \{1,\ldots,n\}$ such that there exists $\Delta \in \Sigma$ such that its m.f.-s.c.s. is $ {\mathbb {R}}^{I'}$.

As $\dim P^{J}<|J|$ for all $J\not \supset I$, we obtain that $I'\supset I$ for all $I'\in \mathcal {S}$. We define

\[ \Sigma(I')=\{\Delta\in\Sigma: \text{the m.f.-s.c.s. of } \Delta \text{ is } {\mathbb{R}}^{I'}\}. \]

Let us consider the set

\[ \Sigma^{I'}:=\{\Delta^{I'}: \Delta\in \Sigma(I')\}=\{\sigma_1,\ldots,\sigma_{l(I')}\}. \]

Given $\sigma _i\in \Sigma ^{I'}$ , let $C_i:=\{\Delta \in \Sigma (I') : \Delta ^I=\sigma _i\}$. Consider the closed set

\[ Z_{(i,I')}:=\bigcup_{\Delta\in C_i}\Delta. \]

Observe that given $\alpha \in \sigma _i^{\circ }$ (where $\sigma _i^{\circ }$ is the relative interior of $\sigma _i$), there exists $\epsilon >0$ such that for each $J\supset I'$, we have $B_{\epsilon }(\alpha )\cap Z_{(i,I')}^J=B_{\epsilon }(\alpha )\cap {\mathbb {R}}_{\geq 0}^J$. Indeed, as $P^J$ is topologically equivalent to a $|J|$-dimensional closed disk for all $J\supset I'$, there exists $\epsilon >0$ such that $B_{\epsilon }(\alpha )\cap {\mathbb {R}}_{\geq 0}^J\subset P^J$. Making $\epsilon$ smaller we may assume that $B_{\epsilon }(\alpha )\cap {\mathbb {R}}_{\geq 0}^J \subset Z_{(i,I')}^{J}$. This implies that $\pi _{I'}\big (Z_{(i,I')}\big )$ is a convenient polyhedron in $ {\mathbb {R}}_{I'}$ (remember that if $(\beta _1,\ldots,\beta _n)\in \operatorname {Ver}(P)$, then for each $i\in I^c$ we have $\beta _i\geq 1$ or $\beta _i=0$), from which it follows that $\nu \big (\pi _{I'}\big (Z_{(i,I')}\big )\big )\geq 0$ (see Theorem 2.4). Now using Proposition 2.1 we obtain $\nu \big (Z_{(i,I')}\big )= |I|!V_{|I|}(\sigma _i)\nu \big (\pi _{I'}\big (Z_{(i,I')}\big )\big )\geq 0$.

By construction, we obtain

\[ P=\bigcup_{I'\in \mathcal{S}}\bigcup_{i=1}^{l(I')} Z_{(i,I')} \]

and

\[ \dim( Z_{(i,I')}^{J'}\cap Z_{(i',I'')}^{J'})<|J'|\quad \text{for all } (i,I')\neq (i',I''). \]

This implies that

\[ \nu(P)=\sum_{I'\in S}\sum_{i=1}^{l(I')}\nu\big(Z_{(i,I')}\big). \]

Rearranging the indices, we obtain the desired subdivision.

Let $S$ and $S'$ be two closed discrete subsets of $ {\mathbb {R}}_{\geq 0}^{n}{\setminus} \{o\}$ such that

\[ \Gamma_{+}(S)\subset\Gamma_{+}(S'). \]

We define $\operatorname {Ver}(S',S):=\operatorname {Ver}(S') {\setminus} \operatorname {Ver}(S)$. The following result tells us where the vertices $\operatorname {Ver}(S',S)$ are found.

Proposition 2.8 Let $S$, $S'$ be two convenient closed discrete subsets of $ {\mathbb {R}}^{n}_{\geq 0}{\setminus} \{o\}$. Suppose that $\Gamma _{+}(S)\subsetneqq \Gamma _{+}(S')$ and $\nu (S)=\nu (S')$. Then

\[ \operatorname{Ver}(S',S)\subset\big({\mathbb{R}}^n_{\geq 0}{\setminus} {\mathbb{R}}^n_{>0}\big). \]

Proof. Let us suppose that $\operatorname {Ver}(S',S)\not \subset ( {\mathbb {R}}^n_{\geq 0}{\setminus} {\mathbb {R}}^n_{>0})$. Let

\[ W= \operatorname{Ver}(S',S)\cap ({\mathbb{R}}^n_{\geq 0}{\setminus} {\mathbb{R}}^n_{>0}) \]

and $\alpha \in \operatorname {Ver}(S',S){\setminus} W$. Let us consider $S'':=S\cup \{\alpha \}$. As the closed discrete sets $S$, $S'$, and $S''$ are convenient and

\[ \Gamma_{+}(S)\subset \Gamma_{+}(S'')\subset\Gamma_{+}(S'), \]

we obtain $\nu (S'')=\nu (S)=\nu (S')$ (see Corollary 2.5). Let us prove that this is a contradiction. In effect, by definition of Newton number we have

\begin{align*} \nu(S)&=n!V_{n} -(n-1)!V_{n-1}+\cdots (-1)^{n-1}V_1+(-1)^{n},\\ \nu(S'')&=n!V''_{n} -(n-1)!V''_{n-1}+\cdots (-1)^{n-1}V''_1+(-1)^{n}, \end{align*}

where $V_{k}:=V_k(\Gamma _{-}(S))$ and $V''_{k}:=V_k(\Gamma _{-}(S''))$ are the ${k}$-dimensional Newton volumes of $\Gamma _{-}(S)$ and $\Gamma _{-}(S'')$, respectively. By construction, $V''_{n}< V_{n}$ and $V'_k=V_k$, $1\leq k\leq n-1$, which implies that $\nu (S'')<\nu (S)$.

If we suppose that $\nu (S')=\nu (S)$, it is not difficult to verify that this equality is not preserved by homotheties of $ {\mathbb {R}}_{\geq 0}^{n}$. The following result describes certain partial homotheties of $ {\mathbb {R}}_{\geq 0}^{n}$ which preserve the equality of the Newton numbers.

Let us consider $D(S,S')=\{I\subset \{1,2,\ldots,n\}:\Gamma _{-}(S)\cap {\mathbb {R}}^{I}\neq \Gamma _{-}(S')\cap {\mathbb {R}}^{I}\}$ and $I(S,S')=\bigcap _{I\in D(S,S')}I$. It may happen that

\[ I(S,S')\notin D(S,S') \]

or

\[ I(S,S')=\emptyset. \]

Proposition 2.9 Let $S, S'\subset {\mathbb {R}}_{\geq 0}^{n}{\setminus} \{o\}$ be two pre-convenient closed discrete sets such that $\Gamma _{+}(S)\subset \Gamma _{+}(S')$. Suppose that $\{1,2,\ldots,k\}\subset I(S,S')$, and consider the map

\[ \varphi_{\lambda}(x_1,\ldots,x_n)=(\lambda x_1,\ldots,\lambda x_k,x_{k+1},\ldots, x_n),\quad \lambda \in {\mathbb{R}}_{>0}. \]

Then $\nu (\varphi _{\lambda }(S'))-\nu (\varphi _{\lambda }(S))=\lambda ^k(\nu (S')-\nu (S))$.

Proof. We use the notation $V_m(S):=V_m(\Gamma _{-}(S))$. Recall that

\[ V_m(S)=\sum_{|I|= m}\operatorname{Vol}_{m}(\Gamma_{-}(S)\cap{\mathbb{R}}^{I}), \]

where $\operatorname {Vol}_{m}(\cdot )$ is the $m$-dimensional volume.

Let $J=\{1,2,\ldots,k\}$. Observe that if $J\not \subset I$, then

\[ \Gamma_{-}(S)\cap{\mathbb{R}}^{I}=\Gamma_{-}(S')\cap{\mathbb{R}}^{I}, \]

which implies that $\operatorname {Vol}_{|I|}(\Gamma _{-}(\varphi _{\lambda }(S))\cap {\mathbb {R}}^{I})=\operatorname {Vol}_{|I|}(\Gamma _{-}(\varphi _{\lambda }(S'))\cap {\mathbb {R}}^{I})$. In particular, if $m< k$ we have $V_m(\varphi _{\lambda }(S))=V_m(\varphi _{\lambda }(S'))$. Let us suppose that $m\geq k$. Then

\[ V_m(\varphi_{\lambda}(S'))-V_m(\varphi_{\lambda}(S))=\sum_{\substack{|I|=m\\ J\subset I}}(\operatorname{Vol}_{m}(\Gamma_{-}(\varphi_{\lambda}(S'))\cap {\mathbb{R}}^{I})-\operatorname{Vol}_{m}(\Gamma_{-}(\varphi_{\lambda}(S))\cap {\mathbb{R}}^{I})). \]

From this we obtain that $V_m(\varphi _{\lambda }(S'))-V_m(\varphi _{\lambda }(S))=\lambda ^k(V_m(S')-V_m(S))$ and $\nu (\varphi _{\lambda }(S'))-\nu (\varphi _{\lambda }(S))=\lambda ^k(\nu (S')-\nu (S))$.

The following corollary is an analogue of Proposition 2.8 in the pre-convenient case. Remember that given $S\subset {\mathbb {R}}_{\geq }^{n}{\setminus} \{o\}$ and $R\subset {\mathbb {R}}^n_{\geq 0}$, we use the notation $S(R):=S\cup R$.

Proof. Observe that there exists $\lambda >0$ such that the closed discrete sets $\varphi _{\lambda }(S)$, $\varphi _{\lambda }(S(\alpha ))$ are convenient where $\varphi _{\lambda }$ is the homothety consisting of multiplication by $\lambda$. As $I(S,S(\alpha ))=\{1,\ldots,n\}$, we have

\[ \nu(\varphi_{\lambda}(S))-\nu(\varphi_{\lambda}(S(\alpha)))=\lambda^n(\nu(S)-\nu(S(\alpha))) \]

(see Proposition 2.9). By Theorem 2.4, we have $\nu (\varphi _{\lambda }(S(\alpha )))\leq \nu (\varphi _{\lambda }(S))$, hence $\nu (S(\alpha ))\leq \nu (S)$. If

\[ \nu(S(\alpha))=\nu(S) \]

then $\nu (\varphi _{\lambda }(S))=\nu (\varphi _{\lambda }(S(\alpha ))$. This contradicts Proposition 2.8.

Take a set $I\subset \{1,\dots,n\}$.

Corollary 2.11 Let $S$, $S'$, and $S''$ be three $I^c$-convenient closed discrete sets such that $\Gamma _{+}(S)\subset \Gamma _{+}(S')\subset \Gamma _{+}(S'')$. Suppose that

\[ I\subset I(S,S')\cap I(S',S'') \]

Then $\nu (S)\geq \nu (S')\geq \nu (S'')$.

Proof. Without loss of generality, we may take $I=\{1,\ldots,k\}$. As $S$, $S'$, and $S''$ are $I^c$-convenient, there exists $\lambda >0$ such that after applying the map $\varphi _{\lambda }$ given by $\varphi _{\lambda }(x_1,\ldots,x_n)=(\lambda x_1,\ldots,\lambda x_k,x_{k+1},\ldots, x_n)$, the closed discrete sets $\varphi _{\lambda }(S)$, $\varphi _{\lambda }(S')$, and $\varphi _{\lambda }(S'')$ are convenient.

As $I\subset I(S,S')\cap I(S',S'')$, we have

\[ \nu(\varphi_{\lambda}(S))-\nu(\varphi_{\lambda}(S'))=\lambda^k(\nu(S)-\nu(S')), \]

and

\[ \nu(\varphi_{\lambda}(S'))-\nu(\varphi_{\lambda}(S''))=\lambda^k(\nu(S)-\nu(S'')). \]

By Theorem 2.4, we obtain $0\leq \nu (S)-\nu (S')$ and $0\leq \nu (S')-\nu (S'')$.

Given $I\subset \{1,2,\ldots,n\}$, let $ {\mathbb {R}}^{I}_{>0}:=\{(x_1,x_2,\ldots,x_n)\in {\mathbb {R}}^{I}:x_i>0\ \text {if}\ i\in I\}$. Let $S\subset {\mathbb {R}}_{\geq }^{n}{\setminus} \{0\}$ be a closed discrete set, and let $\alpha \in {\mathbb {R}}_{>0}^{I}$ be such that

\[ \Gamma_{+}(S)\subsetneqq\Gamma_{+}(S(\alpha)). \]

Let $E_{\alpha }$ be an edge of $\Gamma (S(\alpha ))$ such that $\alpha$ is one of its endpoints. Given a set $J$ with

\[ I\subsetneqq J\subset \{1,2,\ldots,n \}, \]

we say that $E_{\alpha }$ is $(I,J)$-convenient if for all

\[ \beta:=(\beta_1,\ldots,\beta_n)\in (E_{\alpha}\cap \operatorname{Ver}(S)) \]

we have $\beta _{i}\geq 1$ for $i\in J{\setminus} I$ and $\beta _i=0$ for $i\in J^{c}$. We say that $E_{\alpha }$ is strictly $(I,J)$-convenient if $E_{\alpha }$ is $(I,J)$-convenient and whenever

\[ \beta\in(E_{\alpha}\cap\operatorname{Ver}(S)), \]

there exists $i\in J{\setminus} I$ such that $\beta _i>1$.

Example 2.12 Let $0< a<1$, $S:=\{(2,0), (0,2), (\frac {3}{2}(1-a), 2a)\}$, and $\alpha := (\frac {3}{2}, 0)$. For the edge $E_{\alpha }$ of $\Gamma _{+}(S(\alpha ))$ (with endpoints $(\frac {3}{2}, 0)$ and $(0,2)$), we have

\[ E_{\alpha}\cap \operatorname{Ver}(S)=\{(0,2), (\tfrac{3}{2}(1-a), 2a)\}. \]

Then $E_{\alpha }$ is $(\{1\}, \{1,2\})$-convenient (respectively, strictly $(\{1\}, \{1,2\})$-convenient) if and only if $\frac {1}{2}\leq a <1$ (respectively, $\frac {1}{2}< a <1$).

Example 2.13 Let $0< a<1$, $S:=\{(1,0,0), (0,2,0), (\frac {3}{4}(1-a), 2a,0), (0,0,1)\}$, and $\alpha := (\frac {3}{4}, 0,0)$. For the edge $E_{\alpha }$ of $\Gamma _{+}(S(\alpha ))$ (with endpoints $(\frac {3}{4}, 0,0)$ and $(0,2,0)$) we have

\[ E_{\alpha}\cap \operatorname{Ver}(S)=\{(0,2,0), (\tfrac{3}{4}(1-a), 2a,0)\}. \]

Then $E_{\alpha }$ is $(\{1\}, \{1,2\})$-convenient (respectively, strictly $(\{1\}, \{1,2\})$-convenient) if and only if $\frac {1}{2}\leq a <1$ (respectively, $\frac {1}{2}< a <1$).

The following proposition allows us to eliminate certain vertices.

Proposition 2.14 Let $S\subset {\mathbb {R}}_{\geq 0}^{n}{\setminus} \{0\}$ be an $I^{c}$-convenient closed discrete set, $J$ a set such that $I\subsetneqq J\subset \{1,\ldots,n\}$, and $\alpha \in {\mathbb {R}}^{I}_{>0}$ such that

\[ \Gamma_{+}(S)\subsetneqq\Gamma_{+}(S(\alpha)) \]

and $\nu (S(\alpha ))=\nu (S)$. Assume that at least one of the following conditions are satisfied:

1. (i) $\alpha '\in \overline {\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}^I$;

2. (ii) $\alpha '\in \overline {\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}_{>0}^J$ and there exists a strictly $(I,J)$-convenient edge $E_{\alpha }$ of $\,\Gamma (S(\alpha ))$.

Then $\nu (S(\alpha '))=\nu (S)$.

Remark 2.15 In Example 2.12 we have $\nu (S)= \nu (S(\alpha ))$ if and only if

\[ a=\tfrac{1}{2}. \]

In particular, if $\nu (S)= \nu (S(\alpha ))$ then there are no strictly $(\{1\}, \{1,2\})$-convenient edges for $\alpha$.

Observe that for each $0< a<1$, if $\alpha ' \in ( \Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S))\cap {\mathbb {R}}^2_{>0}$, then $\mu (S)\neq \mu (S(\alpha '))$.

In Example 2.13 we have $\nu (S)= \nu (S(\alpha ))$ for all $0< a<1$. Furthermore, for all $\alpha '\in \overline {\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}^{\{1,2\}}$ we have $\nu (S(\alpha '))=\nu (S)$, which indicates that the hypotheses of the preceding proposition are just sufficient.

Observe that $\overline {\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}^{\{1,2\}}$ is the region of the plane bounded by the triangle of the vertices $(\frac {3}{4},0,0)$,$(2,0,0)$ and $(\frac {3}{4}(1-a), 2a,0)$).

Proof. Let us assume that $\alpha '\in \overline {\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}^{I}$. We may assume that $\Gamma _{+}(S)\subsetneqq \Gamma _{+}(S(\alpha '))\subsetneqq \Gamma _{+}(S(\alpha ))$ (otherwise there is nothing to prove).

Observe that the closed discrete sets $S$, $S(\alpha ')$, and $S(\alpha )$ are $I^c$-convenient and

\[ I\subset I(S,S(\alpha'))\cap I(S(\alpha'),S(\alpha)). \]

Using Corollary 2.11, we obtain $\nu (S)=\nu (S(\alpha '))=\nu (S(\alpha ))$. This completes the proof in case (i).

Next, assume that case (ii) holds. Consider a strictly $(I,J)$-convenient edge $E_{\alpha }$ of $\Gamma (S(\alpha ))$. Let $\beta :=(\beta _1,\ldots,\beta _n)\in E_{\alpha }\cap \operatorname {Ver}(S)$. Let $E'\subset E_{\alpha }$ be the line segment with endpoints $\alpha$ and $\beta$. Without loss of generality, we may assume that $E'\cap \operatorname {Ver}(S)=\{\beta \}$. As $E_{\alpha }$ is strictly $(I,J)$-convenient, there exists $i\in J{\setminus} I$ such that $\beta _i>1$. Let $\delta >0$ be sufficiently small so that $\beta _i-\delta \geq 1$ and let $\beta '\in {\mathbb {R}}_{\geq 0}^I$ be such that $\gamma :=\beta -\delta e_i+\beta '\in \Gamma (S(\alpha ))\cap {\mathbb {R}}^J_{>0}$. Then

\[ \Gamma_{+}(S)\subsetneqq\Gamma_{+}(S(\gamma))\subsetneqq\Gamma_{+}(S(\alpha)). \]

Observe that the closed discrete sets $S$s, $S(\gamma )$, and $S(\alpha )$ are $I^c$-convenient and

\[ I\subset I(S,S(\gamma))\cap I(S(\gamma),S(\alpha)). \]

Then $\nu (S(\gamma ))=\nu (S(\alpha ))=\nu (S)$.

If $\alpha '\in \overline { \Gamma _{+}(S(\gamma )){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}^{J}$, we have

\[ \Gamma_{+}(S)\subset \Gamma_{+}(S(\alpha'))\subset \Gamma_{+}(S(\gamma)). \]

The closed discrete sets $S$, $S(\alpha ')$, and $S(\gamma )$ are $J^{c}$-convenient and

\[ J\subset I(S,S(\alpha'))\cap I(S(\alpha'),S(\gamma)). \]

Then $\nu (S(\alpha '))=\nu (S(\gamma ))=\nu (S)$.

We still need to study the case $\alpha '\in (\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S(\gamma )))\cap {\mathbb {R}}^J_{>0}$.

Consider the compact set $C:=\overline {(\Gamma _{+}(S(\alpha ')){\setminus} \Gamma _{+}(S))}\cap {\mathbb {R}}^J$, and the map

\[ \nu_S:C\rightarrow {\mathbb{R}};\;\tau\mapsto \nu_S(\tau):= \nu(S(\tau))=\sum_{m=0}^{n}(-1)^{n-m}m!V_m(S(\tau)) \]

where $V_m(S(\tau )):=V_m(\Gamma _{-}(S(\tau ))))$. The map $\nu _S$ is continuous in $C$. In effect, recall that $V_m(S(\tau ))=\sum _{|I'|= m}\operatorname {Vol}_{m}(\Gamma _{-}(S(\tau ))\cap {\mathbb {R}}^{I'})$. Hence,

\[ V_m(S(\tau))=V(\tau)+V'(\tau), \]

where

\[ V(\tau):= \sum_{\substack{|I'|=m\\ I'\supseteq J}}\operatorname{Vol}_{m}(\Gamma_{-}(S(\tau))\cap {\mathbb{R}}^{I'}),\quad V'(\tau):= \sum_{\substack{ |I'|=m\\ I'\not\supset J}}\operatorname{Vol}_{m}(\Gamma_{-}(S(\tau))\cap {\mathbb{R}}^{I'}). \]

The function $V: {\mathbb {R}}^J\rightarrow {\mathbb {R}};\tau \mapsto V(\tau )$ is continuous, because each summand is continuous in $ {\mathbb {R}}^J$. The function $V':C\rightarrow {\mathbb {R}};\tau \mapsto V'(\tau )$ is constant, because $\Gamma _{-}(S(\alpha '))\cap ( {\mathbb {R}}_{\geq 0}^J{\setminus} {\mathbb {R}}_{>0}^J) =\Gamma _{-}(S) \cap ( {\mathbb {R}}_{\geq 0}^J{\setminus} {\mathbb {R}}_{>0}^J)$. Then each $V_{m}(S(\tau ))$ is continuous in $\tau \in C$, which implies that the function $\nu _S$ is continuous in $C$.

Let us assume that $\alpha '\in (\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S(\gamma )))\cap {\mathbb {R}}^J_{>0}$ and $\alpha '\notin \Gamma (S(\alpha ))$. Let us suppose that $\nu _S(\alpha ')=\nu (S(\alpha '))\neq \nu (S)$. Let us consider the set $\mathcal {C}:=\{ \tau \in C:\nu _S(\tau ))=\nu _S(\alpha '))\}$. The continuity of $\nu _S$ implies that $\mathcal {C}$ is compact. We define the following partial order on $\mathcal {C}$. For $\tau,\tau '\in \mathcal {C}$ we will say that $\tau \leq \tau '$ if $\Gamma _{+}(S(\tau '))\subset \Gamma _{+}(S(\tau ))$. Let us consider an ascending chain

\[ \tau_1\leq \tau_2\leq \cdots \leq \tau_n\leq \cdots. \]

We prove that this chain is bounded above in $\mathcal {C}$. Let us consider the convex closed set

\[ \Gamma=\bigcap_{i\geq 1} \Gamma_{+}(S(\tau_i)). \]

As $\mathcal {C}$ is compact, the sequence $\{\tau _1,\tau _2,\ldots,\tau _n,\ldots \}$ has a convergent subsequence $\{\tau _{i_1}, \tau _{i_2},\ldots,\tau _{i_n},\ldots \}$. Observe that

\[ \Gamma_{+}(S(\tau))=\bigcap_{n\geq 1} \Gamma_{+}(S(\tau_{i_n})), \]

where $\tau :=\lim _{n\rightarrow \infty }\tau _{i_n}\in \mathcal {C}$. By definition, $\Gamma \subset \Gamma _{+}(S(\tau ))$, and by construction for each $i\geq 1$ there exists $n\geq 1$ such that $\Gamma _{+}(S(\tau _{i_n}))\subset \Gamma _{+}(S(\tau _{i}))$. Then $\Gamma =\Gamma _{+}(S(\tau ))$, which implies that $\tau _i\leq \tau$ for all $i\geq 1$. By Zorn's lemma $\mathcal {C}$ contains at least one maximal element. Let $\tau \in \mathcal {C}$ be a maximal element. Recall that we consider $\alpha '\notin \Gamma (S(\alpha ))$, and we made the assumption that $\nu (S(\alpha '))\neq \nu (S)$. Hence, $\tau \notin (\overline { \Gamma _{+}(S(\gamma )){\setminus} \Gamma _{+}(S)})\cap {\mathbb {R}}^{J}$.

Observe that for all $\alpha ''\in \Gamma _{+}(S(\alpha ))$ we have

\[ \Gamma_{+}(S)\subset \Gamma_{+}(S(\alpha''))\subset \Gamma_{+}(S(\gamma,\alpha''))\subset\Gamma_{+}(S(\alpha)). \]

As the closed discrete sets $S$, $S(\alpha '')$, and $S(\gamma,\alpha '')$ are $I^{c}$-convenient and

\[ I\subset I(S(\alpha''),S(\gamma,\alpha''))\cap I(S(\gamma,\alpha''), S(\alpha)), \]

we obtain $\nu (S(\gamma,\alpha ''))=\nu (S(\alpha ''))=\nu (S)$.

As $\tau \notin \overline {\Gamma _{+}(S(\gamma )){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}^{J}$ and $\gamma \in \Gamma (S(\alpha ))$, there exists a relatively open subset $\Omega$ of the relative interior of $\overline {\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}^{I}$ such that $\tau$ belongs to the relative interior of

\[ (\Gamma_{+}(S(\gamma, \alpha'')){\setminus} \Gamma_{+}(S(\alpha''))\cap R^J \]

for all $\alpha ''\in \Omega$. We obtain

\[ \Gamma_{+}(S(\alpha''))\subsetneqq\Gamma_{+}(S(\tau,\alpha''))\subsetneqq \Gamma_{+}(S(\gamma,\alpha'')). \]

The closed discrete sets $S(\alpha '')$, $S(\tau,\alpha '')$, and $S(\gamma,\alpha '')$ are $J^{c}$-convenient, and

\[ J\subset I(S(\alpha''),S(\tau,\alpha''))\cap I(S(\tau,\alpha''),S(\gamma,\alpha'')). \]

Hence, $\nu (S(\tau,\alpha ''))=\nu (S(\alpha ''))=\nu (S)$.

Given an edge $E_{\tau }$ of $\Gamma (S(\tau ))$ that connects $\tau$ with a vertex in $\operatorname {Ver}(\Gamma (S))$, let $E'_{\tau }$ be the subsegment of $E_{\tau }$ containing $\tau$ such that $|E'_{\tau }\cap \operatorname {Ver}(S)|=1$. We choose $\alpha ''\in \Omega '$ such that for each edge $E_{\tau }$ of $\Gamma (S(\tau ))$ connecting $\tau$ with an element of $\operatorname {Ver}(\Gamma (S))$ we have $\dim (E'_{\tau }\cap \Gamma (S(\alpha '')))=0$. In other words, no subsegment of $E'_{\tau }$ is contained in the Newton boundary $\Gamma (S(\alpha ''))$.

Let us consider the compact polyhedron $P:=\overline {\Gamma _{+}(S(\tau,\alpha '')){\setminus} \Gamma _{+}(S(\alpha ''))}$. Observe that $\nu (P)=0$ (see Theorem 2.4).

Given the choice of $\alpha ''$, there exists $\tau '\in P$ such that

\[ \Gamma_{+}(S(\tau'))\subsetneqq\Gamma_{+}(S(\tau)) \]

and $Q_0:=\overline {(\Gamma _{+}(S(\tau )){\setminus} \Gamma _{+}(S(\tau ')))}\subset P$ (it is for achieving the last inclusion that the choice of $\alpha ''$ is really important).

Let $Q_1:=\overline {P{\setminus} Q_0}$. As $\dim (Q_0^{J'}\cap Q_{1}^{J'})<|J'|$, for all $J'\subset \{1,\ldots,n\}$, we obtain $\nu (P)=\nu (Q_0)+\nu (Q_1)$. The polyhedra $ {Q}_0$ and $ {Q}_1$ satisfy the hypotheses of Proposition 2.6. In effect, we have the following.

1. (i) By construction $Q_0$ and $Q_1$ are pure $n$-dimensional compact polyhedra and $o\not \in P=Q_0\cup Q_1$.

2. (ii) Recall that $\tau \in {\mathbb {R}}^J_{>0}$. The polyhedron $P$ satisfies

   \[ \dim(P^{J'})<|J'|\quad\text{for all }J'\not \supset J, \]
   which implies $\dim (Q_0^{J'})<|J'|$ and $\dim (Q_1^{J'})<|J'|$ for all $J'\not \supset J$.

3. (iii) Now we verify that $Q_{0}^{J'}$ is homeomorphic to a $|J'|$-dimensional closed disk for all $J'\supset J$. As $S$ is $I^c$-convenient and $\tau \in {\mathbb {R}}^J_{>0}$, we have $\dim (Q_{0}^{J'})=|J'|$ for each $J'\supset J$. By Remark 2.7 we obtain that $Q_{0}^{J'}$ is homeomorphic to a $|J'|$-dimensional closed disc. The proof for $Q_{1}^{J'}$ is analogous to the proof for $Q_{0}^{J'}$.

4. (iv) As $S$ is $I^c$-convenient (in particular, $J^c$-convenient), we obtain that if $(\beta _1,\ldots,\beta _n)\in \operatorname {Ver}(P)$, then for each $i\in J^c$ we have $\beta _i\geq 1$ or $\beta _i=0$. This property is inherited by $Q_0$ and $Q_1$.

By Proposition 2.6 we have $\nu (Q_0)\geq 0$, $\nu (Q_1)\geq 0$. As $\nu (P)=0$, we obtain $\nu (Q_0)=\nu (Q_1)\!=\! 0$. We have $\tau <\tau '\in \mathcal {C}$, which contradicts the maximality of $\tau$ in $\mathcal {C}$. As a consequence, we obtain $\nu (S(\alpha '))=\nu (S)$.

Now let us suppose that $\alpha ' \in \Gamma (S(\alpha ))\cap {\mathbb {R}}_{>0}^J$, and let $v\in {\mathbb {R}}^J_{>0}$. For $\epsilon >0$ small enough $\alpha _{\epsilon }:=\alpha '+\epsilon v$ belongs to the relative interior of $\Gamma _{+}(S(\alpha ')){\setminus} \Gamma _{+}(S)$. For the continuity of $\nu _S$ in $C:=\overline {(\Gamma _{+}(S(\alpha ')){\setminus} \Gamma _{+}(S))}\cap {\mathbb {R}}^J$ we obtain

\[ \lim_{\epsilon \rightarrow 0}\nu_S(\alpha_{\epsilon})=\nu(S(\alpha')), \]

which implies that $\nu (S(\alpha '))=\nu (S)$.

Corollary 2.16 Let $I\subsetneqq J:= \{1,\ldots,n\}$. Let $S, S'\subset {\mathbb {R}}_{\geq 0}^{n}{\setminus} \{o\}$ be two convenient closed discrete sets such that $\Gamma _{+}(S)\subsetneqq \Gamma _{+}(S')$ and $\nu (S)=\nu (S')$. Suppose that there exists $\alpha \in \operatorname {Ver}(S',S)\cap {\mathbb {R}}^I_{>0}$ and an edge $E_{\alpha }$ of $\Gamma (S)$ that is $(I,J)$-convenient. Then there exists

\[ (\beta_1,\ldots,\beta_n)\in \operatorname{Ver}(S)\cap E_{\alpha} \]

such that $\beta _i=1$ for all $i\in I^c$.

Proof. Let $R:= \operatorname {Ver}(S',S){\setminus} \{\alpha \}$ and $S(R)=S\cup R$. The closed discrete sets $S,$ $S(R)$ and $S'$ are convenient, and $\Gamma _{+}(S)\subset \Gamma _{+}(S(R))\subsetneqq \Gamma _{+}(S')$. Then

\[ \nu(S(R))=\nu(S'). \]

We argue by contradiction. If there is no $(\beta _1,\ldots,\beta _n)$ as in the corollary, then the edge $E_{\alpha }$ is strictly $(I,J)$-convenient. By Proposition 2.14, for all $\alpha '\in \overline {\Gamma _{+}(S(\alpha ){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}_{>0}^n$ we have

\[ \nu(S(R))=\nu(S(R\cup\{\alpha'\})), \]

which contradicts Proposition 2.8.

The following proposition allows us to fix a special coordinate hyperplane and gives information about the edges not contained in the hyperplane that contain a vertex of interest belonging to the hyperplane.

Proposition 2.17 Let $S,S'\subset {\mathbb {R}}_{\geq 0}^{n}{\setminus} \{o\}$ be two convenient closed discrete sets such that $\Gamma _{+}(S)\subsetneqq \Gamma _{+}(S')$ and $\nu (S)=\nu (S')$. Let us suppose that

\[ \alpha\in \operatorname{Ver}(S',S)\cap {\mathbb{R}}^I_{>0},\quad I\subsetneqq\{1,\ldots,n\}. \]

Then there exists $i\in I^c$ such that for all the edges $E_{\alpha }$ of $\Gamma (S')$ not contained in $ {\mathbb {R}}_{\{i\}}$ there exists $(\beta _1,\ldots,\beta _n)\in \operatorname {Ver}(S)\cap E_{\alpha }$ such that $\beta _i=1$.

Lemma 2.18 Let $S\subset {\mathbb {R}}_{\geq 0}^{n} {\setminus} \{o\}$ be an $I^c$-convenient closed discrete set and

\[ \alpha\in{\mathbb{R}}^I_{>0},\quad I\subset\{1,\ldots,n\}, \]

such that $\nu (S)=\nu (S(\alpha ))$. Then there exists $i\in I^c$ such that for each edge $E_{\alpha }$ of $\Gamma (S(\alpha ))$ not contained in $ {\mathbb {R}}_{\{i\}}$ there exists $(\beta _1,\ldots,\beta _n)\in \operatorname {Ver}(S)\cap E_{\alpha }$ such that $\beta _i=1$.

Proof of Lemma 2.18. By Corollary 2.10, we have $|I|< n$. Let $k$ be the greatest element of $\{1,\dots,n-1\}$ such that the lemma is false for some $I$ with $|I|=k$. In other words, for all $i\in I^c$ there exists an edge $E_{\alpha }$, not contained in $ {\mathbb {R}}_{\{i\}}$, such that for all $(\beta _1,\ldots,\beta _n)\in \operatorname {Ver}(S)\cap E_{\alpha }$ we have $\beta _i>1$. Let $J\subset \{1,\ldots,n\}$ be a set of the smallest cardinality such that $E_{\alpha }\subset {\mathbb {R}}^J$. Then $E_{\alpha }$ is a strictly $(I,J)$-convenient edge. Using Proposition 2.14 we obtain that for all $\alpha '\in \overline {\Gamma _{+}(S(\alpha ){\setminus} \Gamma _{+}(S)}\cap {\mathbb {R}}_{>0}^J$ we have $\nu (S(\alpha '))=\nu (S)$. Now let us choose $\alpha '$ sufficiently close to $\alpha$ so that for each edge $E_{\alpha '}$ of $\Gamma (S(\alpha '))$, and $\beta \in E_{\alpha '}\cap \operatorname {Ver}(S)$ adjacent to $\alpha '$ in $E_{\alpha '}$, there exists an edge $E_{\alpha }$ of $\Gamma (S(\alpha ))$ such that $\beta \in E_{\alpha }$. Then the closed discrete sets $S$, $S(\alpha ')$ are $J^c$-convenient and do not satisfy the conclusion of the lemma, which is a contradiction, because $|J|>k$.

The proof of the proposition is by induction on the cardinality of $\operatorname {Ver}(S',S)$. Lemma 2.18 says that the proposition is true whenever $|\!\operatorname {Ver}(S',S)|=1$. Let us assume that the proposition is true for all $S$, $S'$ such that

\[ |\!\operatorname{Ver}(S',S)|\leq m-1. \]

Let $S$, $S'$ with $|\!\operatorname {Ver}(S',S)|= m\geq 2$ be such that the proposition is false. Then there exists $\alpha \in \operatorname {Ver}(S',S)$ such that for each $i\in I^c$ there exists an edge $E_{\alpha }$ of $\Gamma _{+}(S')$, not contained in $ {\mathbb {R}}_{\{i\}}$, that satisfies the following condition:

1. () for all $\beta =(\beta _1,\ldots,\beta _n)\in \operatorname {Ver}(S)\cap E_{\alpha }$ we have $\beta _i>1$;

note that condition $(*)$ is vacuously true if

(1)\begin{equation} \operatorname{Ver}(S)\cap E_{\alpha}=\emptyset.\end{equation}

Observe that for each $\alpha '\in \operatorname {Ver}(S',S){\setminus} \{\alpha \}$, we have $|\!\operatorname {Ver}(S',S(\alpha '))|=m-1$ and, by Corollary 2.11, $\nu (S(\alpha '))=\nu (S')$.

First, let us suppose that there exists $i\in I^c$ such that (1) does not hold for the corresponding edge $E_\alpha$. Let us fix $\alpha '\in \operatorname {Ver}(S',S){\setminus} \{\alpha \}$. Then $E_\alpha$ connects $\alpha$ with a vertex $\beta$ of $S$, hence $\alpha '\notin E'$, where $E'\subset E_{\alpha }$ is the line segment with endpoints $\alpha$ and $\beta$. We obtain that the polyhedra $\Gamma _{+}(S(\alpha '))\subsetneqq \Gamma _{+}(S')$ do not satisfy the conclusion of the Proposition, which contradicts the induction hypothesis.

Next, let us suppose that there exists $i\in I^c$ such that (1) is satisfied for the corresponding edge $E_\alpha$. Then $|E_{\alpha }\cap \operatorname {Ver}(S',S)|=2$. Now, take

\[ \alpha''=(\alpha'_1,\ldots,\alpha'_n)\in E_{\alpha}\cap\operatorname{Ver}(S',S) \]

such that $\alpha '\neq \alpha$. If $\alpha '_i>1$, then the Newton polyhedra

\[ \Gamma_{+}(S(\alpha'))\subsetneqq\Gamma_{+}(S') \]

do not satisfy the Proposition and (1) does not hold, which is a contradiction. Hence, $\alpha '_i=1$. Let $\epsilon >0$ be such that

\[ \alpha'_{\epsilon}:=\alpha'+\epsilon e_i\in(\Gamma_{+}(S'){\setminus}\Gamma_{+}(S)). \]

Put

\[ R:=(\operatorname{Ver}(S',S){\setminus}\{\alpha'\})\cup\{\alpha'_{\epsilon}\}. \]

Then $\Gamma _{+}(S)\subsetneqq \Gamma _{+}(S(\alpha '_{\epsilon }))) \subsetneqq \Gamma _{+}(S(R)))\subsetneqq \Gamma _{+}(S')$.

The closed discrete sets $S$, $S(\alpha '_{\epsilon })$, $S(R)$, and $S'$ are convenient. We have

\[ \nu(S(\alpha'_{\epsilon}))=\nu(S(R))=\nu(S). \]

Let us assume that $\epsilon$ is small enough so that there exists an edge $E'_{\alpha }\ni \alpha$ of $\Gamma (S(R))$ such that $\alpha '_{\epsilon }\in E'_{\alpha }$. Then the Newton polyhedra $\Gamma _{+}(S) \subsetneqq \Gamma _{+}(S(R))$ satisfy the preceding case (namely, $\alpha '_i>1$). This completes the proof of the proposition.

Corollary 2.19 Assume given two convenient closed discrete sets

\[ S, S'\subset {\mathbb{R}}_{\geq 0}^{n}{\setminus}\{o\} \]

such that $\Gamma _{+}(S)\subsetneqq \Gamma _{+}(S')$ and $\nu (S)=\nu (S')$. Assume that

\[ \alpha\in \operatorname{Ver}(S',S)\cap {\mathbb{R}}^I_{>0}. \]

Then, for the $i\in I^c$ of Proposition 2.17 there exists an edge $E_{\alpha }$ of $\Gamma (S')$, and $(\beta _1,\ldots,\beta _n)\in E_{\alpha }\cap \operatorname {Ver}(S)$, such that $\beta _j=\delta _{ij}$, $j\in I^c$, where $\delta _{ij}$ is the Kronecker delta.

Proof of the corollary By Proposition 2.17 there exists $i\in I^c$ such that for all the edges $E_{\alpha }$ of $\Gamma (S')$, not contained in $ {\mathbb {R}}_{\{i\}}$, there exists

\[ (\beta_1,\ldots,\beta_n)\in \operatorname{Ver}(S)\cap E_{\alpha} \]

such that $\beta _i=1$. As the set $S$ is convenient, there exists $m>1$ such that $me_i\in \operatorname {Ver}(S)$. Let $J=I\cup \{i\}$. As $\alpha,me_i\in {\mathbb {R}}^J$, there exists a chain of edges of $\Gamma (S')$ connecting $\alpha$ with $me_i$, contained in $ {\mathbb {R}}^J$. The edge $E_\alpha$ belonging to this chain and containing $\alpha$ satisfies the conclusion of the corollary.

The following theorem generalizes to all dimensions the main theorem of [Reference Brzostowski, Krasiński and WalewskaBKW19]. In [Reference Brzostowski, Krasiński and WalewskaBKW19] this result is conjectured.

Definition 2.21 Let $S, S'\subset {\mathbb {R}}_{\geq 0}^{n}{\setminus} \{o\}$ be two closed discrete sets such that

\[ \Gamma_{+}(S)\subsetneqq\Gamma_{+}(S'), \]

$I\subsetneqq \{1,\ldots,n\}$ and $\alpha \in \operatorname {Ver}(S',S)\cap {\mathbb {R}}_{>0}^{I}$. We say that $\alpha$ has an apex if there exists $i\in I^c$ and a unique edge $E_{\alpha }$ of $\Gamma (S')$ that contains $\alpha$, and is not contained in $ {\mathbb {R}}_{\{i\}}$.

In this case the point $\beta \in \operatorname {Ver}(S)\cap E_{\alpha }$ adjacent to $\alpha$ in $E_{\alpha }$ is called the apex of $\alpha$. We say that an apex, $\beta :=(\beta _1,\ldots,\beta _n)$, is good if $\beta _j=\delta _{ij}$, $j\in I^c$.

Remark 2.22 Let $S\subset {\mathbb {R}}_{\geq 0}^{n}{\setminus} \{o\}$ be a convenient closed discrete set,

\[ I\subsetneqq\{1,\ldots,n\} \]

and $\alpha \in {\mathbb {R}}_{>0}^I$ such that $\Gamma _{+}(S)\subsetneqq \Gamma _{+}(S(\alpha ))$. The condition that $\alpha$ has a good apex $\beta \in {\mathbb {R}}^{I\cup \{i\}}_{>0}$, $i\in I^c$, is equivalent to $P:=\overline {\Gamma _{+}(S(\alpha )){\setminus} \Gamma _{+}(S)}$ being a pyramid with apex $\beta$ and base $P\cap {\mathbb {R}}_{\{i\}}$.

Example 2.23 For example, in the case of the $\mu$-constant deformation of Briançon–Speder convenient version (see [Reference Briançon and SpederBS75]),

\[ F(x,y,z,s):= x^{5}+y^{7}z+z^{15}+y^{8}+sxy^{6}, \]

the pyramid is formed by the base with vertices $(5,0,0)$, $(0,8,0)$, $\alpha =(1,6,0)$, and the good apex $\beta =(0,7,1)$. Note that the vertices $(5,0,0)$, $(0,7,1)$, $(1,6,0)$, $(0,0,15)$ are coplanar.

Lemma 2.26 Let $S\subset {\mathbb {R}}_{\geq 0}^{n}{\setminus} \{o\}$ be a closed discrete set and $\alpha \in {\mathbb {R}}_{>0}^I$, $I\subsetneqq \{1,\ldots,n\}$, such that $\Gamma _{+}(S)\subsetneqq \Gamma _{+}(S(\alpha ))$, and $\alpha$ has a good apex. Then

\[ \nu(S(\alpha))=\nu(S). \]

Proof of Lemma 2.26 Let $\beta$ be a good apex of $\alpha$. Let $i\in I^c$ be such that $\beta \in {\mathbb {R}}_{>0}^{I\cup \{i\}}$.

Given an element $m\in \{1,\dots,n\}$ and $J\subset \{1,\ldots,n\}$ such that $|J|=m$, we use the notation

\[ V_m(\alpha, J)=\operatorname{Vol}_{m}(\Gamma_{-}(S(\alpha))\cap {\mathbb{R}}^{J})-\operatorname{Vol}_{m}(\Gamma_{-}(S)\cap {\mathbb{R}}^{J}). \]

As $\alpha \in {\mathbb {R}}^{I}_{>0}$, we have

\begin{align*} \nu(S(\alpha))-\nu(S)&= \sum_{m=|I|}^{n}(-1)^{m} \sum _{\substack{|J|=m\\ I\subset J}}|J|!V_{m}(\alpha,J)\\ &= \sum_{m=|I|}^{n-1}(-1)^{m}\sum_{\substack{|J|=m\\ i\notin J, I\subset J}} (|J|!V_{m}(\alpha,J)-(|J|+1)!V_{m+1}(\alpha,J\cup\{i\})). \end{align*}

As the apex of $\alpha$ is good, we obtain

\[ |J|!V_{m}(\alpha,J)=(|J|+1)!V_{m+1}(\alpha,J\cup\{i\}), \]

which implies that $\nu (S)=\nu (S(\alpha ))$.

Now we prove that if each $\alpha \in \operatorname {Ver}(S',S)$ has a good apex, then

\[ \nu(S)=\nu(S'). \]

The proof is by induction on the cardinality of $\operatorname {Ver}(S',S)$. Let us assume that the implication is true for all $S$ and $S'$ such that $|\!\operatorname {Ver}(S',S)|< m$. To verify the implication for $|\!\operatorname {Ver}(S',S)|=m$, let $\alpha \in \operatorname {Ver}(S',S)$ and $R=\operatorname {Ver}(S',S){\setminus} \{\alpha \}$. By the induction hypothesis $\nu (S(R))=\nu (S)$ and by Lemma 2.26 we have $\nu (S')=\nu (S(R))$. This proves that $\nu (S')=\nu (S)$.

To finish the proof of the theorem we need the following lemma.

Proof of Lemma 2.27 Let $i\in I^c$ be as in Remark 2.20. Then there exists $E_{\alpha }$ of $\Gamma (S(\alpha ))$, and $\beta :=(\beta _1,\ldots,\beta _n)\in E_{\alpha }\cap \operatorname {Ver}(S)$, such that $\beta _j=\delta _{ij}$, $j\in I^c$. We want to prove that $\beta$ is a (necessarily good) apex of $\alpha$. Let us assume that $\beta$ is not an apex of $\alpha$, aiming for contradiction. Then there exits another edge $\alpha \in E'_{\alpha }$ de $\Gamma (S(\alpha ))$, and $\beta ':=(\beta '_1,\ldots,\beta '_n)\in E'_{\alpha }\cap \operatorname {Ver}(S)$ adjacent to $\alpha$ in $E'_{\alpha }$ such that $\beta '_i=1$.

Let us consider $\beta '_{\epsilon }:= \beta '+\epsilon e_i$, and the closed discrete set $S^{\epsilon }=(S{\setminus} \{\beta '\})\cup \{\beta '_{\epsilon }\}$, $\epsilon > 0$. Let us assume that $\epsilon$ is small enough so that:

1. (i) $\operatorname {Ver}(S^{\epsilon })=(\operatorname {Ver}(S){\setminus} \{\beta '\})\cup \{\beta '_{\epsilon }\}$;

2. (ii) there exists an edge $E_{\alpha }^{\epsilon }$ of $\Gamma (S^{\epsilon }(\alpha ))$ such that $\beta _{\epsilon }\in E_{\alpha }^{\epsilon }\cap \operatorname {Ver}(S^{\epsilon })$ is adjacent to $\alpha$ in $E_{\alpha }^{\epsilon }$.

Let $P^{\epsilon }=\overline {(\Gamma _{+}(S^{\epsilon }(\alpha ){\setminus} \Gamma _{+}(S^{\epsilon }))}$. Let $Q_{0}$ be the convex hull of the set

\[ \{\beta\}\cup( P^{\epsilon}\cap{\mathbb{R}}_{\{i\}}) \]

(observe that $Q_0$ does not depend on $\epsilon$) and $Q^{\epsilon }_1:=\overline {P^{\epsilon }{\setminus} Q_0}$. Recall that $\beta$ satisfies $\beta _j=\delta _{ij}$, $j\in I^c$. Then, using the same idea as in the proof of Lemma 2.26 we obtain $\nu (Q_0)=0$. As $\dim (Q_{0}^{J}\cap (Q^{\epsilon }_1)^{J} )<|J|$ for all $J\subset \{1,\ldots,n\}$, we have

\[ \nu(P^{\epsilon})=\nu(Q_0)+\nu(Q^{\epsilon}_1).\ \text{Then } \nu(P^{\epsilon})=\nu(Q^{\epsilon}_1). \]

As $S^{\epsilon }(\alpha )$ is $I^c$-convenient, $Q^{\epsilon }_1$ satisfies the hypotheses of Proposition 2.6 (to prove this statement use the same idea as in the proof of Proposition 2.14). Let us consider the sequence

\[ I\cup\{i\}\subset I_1,I_2,\ldots,I_m\subset \{1,\ldots,n\}, \]

and the polyhedra $Z^{\epsilon }_{j}$, $1\leq j\leq m$, such that:

1. (i) $Q^{\epsilon }_1=\bigcup _{j=1}^{m}Z^{\epsilon }_{j}$;

2. (ii) $\nu (Q^{\epsilon }_1)=\sum _{j=1}^{m}\nu (Z^{\epsilon }_{j})$;

3. (iii) $\nu (Z^{\epsilon }_{j})=|I_{j}|!V_{|I_{j}|}((Z^{\epsilon }_{j})^{I_{j}}) \nu (\pi _{I_{j}}(Z^{\epsilon }_{j}))\geq 0$;

(the existence of these objects is given by Proposition 2.6). For each $j$, $1\le j\le m$, we may choose the family $Z_j^\epsilon$ of polyhedra to vary continuously with $\epsilon$. More precisely, we can choose the $Z_j^\epsilon$ to satisfy the following additional condition: for each $j$, $1\le j\le m$, either $Z_j^\epsilon =Z_j^0$ for all small $\epsilon$ or $\operatorname {Ver}(Z_j^\epsilon )$ differs from $\operatorname {Ver}(Z_j^0)$ in exactly one element, $\beta '_\epsilon \ne \beta '$, for all small $\epsilon >0$. As $i\in I_j$, we have $\pi _{I_j}( \beta '_{\epsilon })=\pi _{I_j}(\beta ')$. This implies that $\nu (\pi _{I_{j}}(Z^{\epsilon }_{j}))$ is independent of $\epsilon$ for all $1\leq {j}\leq m$. For $\epsilon =0$, we have

\[ \nu(\pi_{I_{j}}(Z^{0}_{j}))=0. \]

Hence, $\nu (P^{\epsilon })=\nu (Q_1^{\epsilon })=0$ for $\epsilon$ small enough. Then there exists a set $J$, $\{i\}\cup I\subset J\subset \{1,2,\ldots,n\}$, such that the edge $E^{\epsilon }_{\alpha }$ is strictly $(I,J)$-convenient. By Proposition 2.14, given $\alpha '\in \overline {\Gamma _{+}(S^{\epsilon }(\alpha )){\setminus} \Gamma _{+}(S^\epsilon )}\cap {\mathbb {R}}_{>0}^J$ we have

\[ \nu(S^{\epsilon}(\alpha'))=\nu(S^{\epsilon}). \]

This proves that $|I|< n-1$: indeed, if $|I|=n-1$, then $\alpha '\in {\mathbb {R}}_{>0}^n$, which contradicts Corollary 2.10.

Let $r$ be the largest element of $\{1,\ldots,n-1\}$ such that the lemma is true for all $I$ such that $|I|>r$. Now let us assume that $|I|=r$. Let us choose $\alpha '$ sufficiently close to $\alpha$ so that for each edge $E_{\alpha '}$ of $\Gamma (S^{\epsilon }(\alpha '))$ and

\[ \beta\in E_{\alpha'}\cap \operatorname{Ver}(S^{\epsilon}) \]

adjacent to $\alpha '$ in $E_{\alpha '}$, there exists an edge $E_{\alpha }$ of $\Gamma (S^{\epsilon }(\alpha ))$ such that $\beta \in E_{\alpha }$. This implies that $\alpha '$ does not have a good apex, which contradicts the choice of $r$, since $|J|>r$. This completes the proof of the lemma.

Now we can finish the proof of the theorem. We prove that if

\[ \nu(S)=\nu(S'), \]

then each $\alpha \in \operatorname {Ver}(S',S)$ has a good apex. The proof is by induction on the cardinality of $\operatorname {Ver}(S',S)$. Lemma 2.27 shows that the implication is true for $|\!\operatorname {Ver}(S',S)|=1$. Let us assume that this is true for every pair $(S,S')$ of convenient closed discrete sets such that $|\!\operatorname {Ver}(S',S)|< m$. Let us prove the result for $|\!\operatorname {Ver}(S',S)|=m$. Let $\alpha \in \operatorname {Ver}(S',S)$, $R=\operatorname {Ver}(S',S){\setminus} \{\alpha \}$ and $\alpha _{\epsilon }=(1+\epsilon )\alpha$, where $\epsilon >0$. Then

\[ \Gamma_{+}(S)\subseteq\Gamma_{+}(S(\alpha_{\epsilon}))\subsetneqq\Gamma_{+}(S(\alpha_{\epsilon})(R))\subseteq\Gamma_{+}(S'). \]

By Corollary 2.5 we have $\nu (S(\alpha _{\epsilon })(R))=\nu (S(\alpha _{\epsilon }))$. Observe that

\[ |\!\operatorname{Ver}(S(\alpha_{\epsilon})(R),S(\alpha_{\epsilon}))|\leq m-1. \]

By the induction hypothesis, each $\alpha '\in R$ has a good apex $\beta \in \operatorname {Ver}(S(\alpha _{\epsilon }))$ for the inclusion $\Gamma _{+}(S(\alpha _{\epsilon }))\subsetneqq \Gamma _{+}(S(\alpha _{\epsilon })(R))$ of Newton polyhedra. As all the non-zero coordinates of $\alpha _\epsilon$ are strictly greater than one, we have $\beta \neq \alpha _{\epsilon }$, so that $\beta \in \operatorname {Ver}(S)$. We take $\epsilon$ small enough so that for every $\alpha '\in R$ every edge $E_{\alpha '}$ of $\Gamma (S(\alpha _{\epsilon })(R))$ that connects $\alpha '$ with a vertex in $\operatorname {Ver}(S)$ is an edge of $\Gamma (S')$. Thus, every $\alpha '\in R$ has a good apex for the inclusion

\[ \Gamma_{+}(S)\subset\Gamma_{+}(S') \]

of Newton polyhedra.

Now it suffices to verify that $\alpha$ has a good apex for the inclusion

(2)\begin{equation} \Gamma_{+}(S)\subset\Gamma_{+}(S')\end{equation}

of Newton polyhedra. Let $\epsilon >0$ and put $R_{\epsilon }:=\{(1+\epsilon )\alpha ': \alpha '\in R\}$. Then

\[ \Gamma_{+}(S)\subset\Gamma_{+}(S(R_{\epsilon}))\subsetneqq\Gamma_{+}(S(R_{\epsilon})(\alpha))\subset \Gamma_{+}(S'). \]

By Corollary 2.5 we have $\nu (S(R_{\epsilon }))=\nu (S(R_{\epsilon })(\alpha ))=\nu (S)$. Observe that $\operatorname {Ver}(S(R_{\epsilon })(\alpha ), S(R_{\epsilon }))=\{\alpha \}$. By Lemma 2.27, $\alpha$ has a good apex

\[ \beta\in\operatorname{Ver}(S(R^{\epsilon})). \]

As every non-zero coordinate of every element of $R_\epsilon$ is strictly greater than one, we have $\beta \notin R_{\epsilon }$, so that $\beta \in \operatorname {Ver}(S)$. Take $\epsilon$ small enough so that every edge $E_{\alpha }$ of $\Gamma (S(R_{\epsilon })(\alpha ))$ that connects $\alpha$ with a vertex in $\operatorname {Ver}(S)$ is an edge of $\Gamma (S')$. Then $\beta$ is a good apex of $\alpha$ for the inclusion (2), as desired. This completes the proof of the theorem.

We end this section by recalling a result that relates the Milnor number to the Newton number.

If the formal power series $g$ is not convenient, we can define the Newton number $\nu (g)$ of $g$ ($\nu (g)$ could be $\infty$) in the following way. Let $\mathcal {E'}\subset \mathcal {E}$ such that there does not exist $m\in {\mathbb {Z}}_{>0}$, such that $me\in \operatorname {Ver}(g)$. We define the Newton number of $g$ as

\[ \nu(g):=\sup_{m\in {\mathbb{Z}}_{>0}}\nu(\operatorname{Supp}(g)\cup \mathcal{E'}_m), \]

where $\mathcal {E'}_m:=\{me:e\in \mathcal {E'}\}$.

Example 2.30 Consider the following families of non-degenerate deformations:

\[ F^{\lambda}(x,y,z,s):= x^{5\lambda}+y^{7\lambda}z+z^{15}+y^{8\lambda}+sx^{\lambda}y^{6\lambda}, \quad \lambda\geq 1. \]

Observe that $F^1$ is a $\mu$-constant deformation of Briançon–Speder (convenient version), see [Reference Briançon and SpederBS75]. By virtue of Theorem 2.28 and Proposition 2.9, for each $\lambda \geq 1$ the deformation $F^{\lambda }$ is $\mu$-constant.