3.1. Milnor arcs at infinity

The idea of the Milnor set was introduced by Milnor [Reference Milnor27] and has been used in many papers ever since, either locally for germs of functions [Reference Aoki, Fukuda and Sun1, Reference Araújo dos Santos, Chen and Tibăr2, Reference Araújo dos Santos and Tibăr4, Reference Bekka6, Reference Szafraniec32, Reference Thom37], or globally for polynomial functions [Reference Araújo dos Santos, Chen and Tibăr3, Reference Dias, Ruas and Tibăr8, Reference Dias, Joita and Tibăr9, Reference Monsalve29, Reference Némethi and Zaharia31, Reference Tibăr38, Reference Tibăr39].

Let $\rho _a : {\mathbb {R}}^2 \rightarrow {\mathbb {R}} _{\geq 0}$, $\rho _a(x,y)=(x-a_1)^2+(y-a_2)^2$ be the square of the Euclidean distance to $a:= (a_1 , a_2) \in {\mathbb {R}}^2$.

Definition 3.1 The Milnor set of $f : {\mathbb {R}} ^2 \rightarrow {\mathbb {R}}$ relative to $\rho _{a}$ is the set of $\rho _a$-nonregular points of $f$, namely

\[ M_a(f) := \{ (x,y) \in {\mathbb{R}}^2 \mid \rho_a \not\pitchfork_{(x,y)} f \}. \]

Equivalently, the Milnor set $M_a(f)$ is the zero set $\{{\rm {Jac}\ } F_{a} =0\}$ considered with reduced structure, where ${\rm {Jac}\ } F_{a}$ denotes the determinant of the Jacobian matrix of the map $F_a:= (f, \rho _a): {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2$.

We may and will assume from now on that, modulo a translation of coordinates, the origin 0 is a point of $\Omega (f)$, and for this point we will use the simplified notations without lower index, such as $M(f)$ etc.

3.2. Clusters of Milnor arcs

By proposition 3.2, the Milnor arcs at infinity do not intersect mutually. It follows that if $C\subset {\mathbb {R}}^{2}$ is some large enough circle centred at the origin, then $M(f) \cap C$ is a finite set of points $\{p_{1}, \ldots, p_{s}\}$. We define the following counterclockwise relation between these pointsFootnote ²: we say that ‘$p_{j}$ is the successor of $p_{k}$,’ or that ‘$p_{k}$ is the antecedent of $p_{j}$,’ if starting from the point $p_{k}$ and moving counterclockwise along the circle $C$ one arrives at the point $p_{j}$ without meeting any other point of the set $M(f) \cap C$.

We also say that $\{p_{1}, \ldots, p_{k}\}$ is a sequence of consecutive points of the set $M(f) \cap C$ if $p_{i+1}$ is the successor of $p_{i}$ for all $i=1, \ldots, k-1$. This relation between the points $M(f) \cap C$ on the circle $C$ allows us to define a similar one among the Milnor arcs at infinity, as follows:

A similar definition of Milnor clusters was given in [Reference Hà and Nguyên18] in the setting of surfaces in ${\mathbb {R}}^{n}$ instead of ${\mathbb {R}}^{2}$. Earlier, polar clusters have been defined in [Reference Coste and de la Puente7]. In [Reference Coste and de la Puente7, Reference Hà and Nguyên18], clusters are used for detecting atypical fibres. An effective detection of atypical values via Milnor clusters can be found in [Reference Monsalve29]. Let us point out that [Reference Dias, Joita and Tibăr9] develops an algorithmic detection of atypical fibres without using Milnor clusters.

Convention. In the rest of this paper we shall designate the Milnor arcs at infinity simply as ‘Milnor arcs.’

3.3. The Splitting (Sp) and the Vanishing (Va) at infinity

The splitting (denoted Sp) and the vanishing (denoted Va) of fibre components are phenomena which may happen, at some point $(p, \lambda )\in {\mathcal {L}}_{f}\times {\mathbb {R}}$ where $\lambda$ denotes a value of $f$. These have been definedFootnote ⁴ in [Reference Tibăr and Zaharia40]. They are related to Milnor arcs at infinity. More precisely, we will see in the following that any odd cluster is either a splitting cluster or a vanishing cluster.

After [Reference Dias, Joita and Tibăr9] (see theorem 3.13 below), the existence of atypical fibres is equivalent to the existence of atypical points at infinity in ${\mathcal {L}}_f \times {\mathbb {R}}$, which are defined with respect to the local splitting and local vanishing (which are the localizations of the Sp and Va phenomena).

In order to display the definitions, let us recall a few preliminaries following [Reference Dias, Joita and Tibăr9]. Let $\{ M_t \}_{t\in {\mathbb {R}}}$ be a family of sets in ${\mathbb {R}}^2$. We say that the limit set of the family $\{ M_t \}_{t\in {\mathbb {R}}}$ when $t \to \lambda$, and we denote it by $\lim _{t\to \lambda } M_t$, is the set of points $x \in {\mathbb {R}}^2$ such that there exists a sequence $t_k \in {\mathbb {R}}$ with $t_k \to \lambda$ and a sequence of points $x_{k} \in M_{t_k}$ such that $x_{k} \to x$.

The paper [Reference Dias, Joita and Tibăr9] shows that one can localize at some points $(p,\lambda ) \in L^{\infty } \times {\mathbb {R}}$ the vanishing and the splitting at infinity of $f$ at $\lambda$ (cf. definition 3.9). To explain this result that we will need here, let us recall some notations and definitions.

By a linear change of coordinates we may assume, without loss of generality, that $p \in L^\infty$ is the point $[0:1:0]$. Recall that $\tilde f (x,y,z)$ denotes the homogenization of degree $d= \deg f$ with respect to the new variable $z$. In some chart $U \simeq {\mathbb {R}}^2 \subset {\mathbb {P}}^2$ at $p$, the family of polynomial functions $g_t:= \tilde f(x,1,z)-tz^d$ defines a family of algebraic curve germs $C_t:=\{g_t =0\}$ at $p$, of parameter $t$.

By comparing the proof of theorem 3.13 to definition 3.12, we immediately get the following rephrasingFootnote ⁵:

However, let us note that corollary 3.14 is not anymore true for odd clusters corresponding to connected components of fibres of $f$ which tend to the values $\pm \infty$, see example 6.4.

3.4. Points at infinity, clusters and tangents

We recall that the notation ${\mathcal {L}}_{f}$ stands for the points at infinity of all the fibres of a non-constant polynomial $f: {\mathbb {R}}^{2}\to {\mathbb {R}}$, and $|{\mathcal {L}}_{f}| = \# {\mathcal {L}}_{f}$. Also recall that a vanishing or a splitting cluster at some value $\lambda$ contains an odd number of Milnor arcs, by theorem 3.7, and that all these arcs contain the point $p$ in their closure at infinity, by corollary 3.14.

4.1. Index of a Milnor arc, after [12]

Recall that the Milnor set $M(f)$ is the set of points where the fibres of $f$ are tangent to the level sets of the Euclidean distance function $\rho$, and that, by definition 3.2, the Milnor arcs do not intersect ${\rm {Sing}\ } f$. For any point $q$ of a Milnor arc $\gamma$ outside a disk $D = \{\rho (x,y) \le R \}$ of large enough radius $R$, the fibre of $f$ passing through $q$ may be in only one of the following three situations:

1. (a) locally inside the disk $D$, and then one defines the index $i(\gamma ) := +\frac {1}{2}$

2. (b) locally outside $D$, and then one defines the index $i(\gamma ) := -\frac {1}{2}$,

3. (c) a local half-branch inside $D$ and the other local half-branch outside $D$, in which case one defines the index $i(\gamma ) := 0$. Actually, we will see in the proof of lemma 4.3 that, for a generic choice of the origin, the Milnor set $M(f)$ does not contain Milnor arcs $\gamma$ of index $0$.

It then follows that along a fixed fibre component, outside the disk $D$, the distance function has alternating local maxima and minima in the counterclockwise order, and without counting the inflexion points. Therefore we get:

Now by theorem 3.7, remark 3.8, and lemma 4.1, we directly get the following consequence:

Let us set the following notation:

\[ i_{p,c} := \sum_{\gamma} i(\gamma) \]

where $\gamma$ runs over all Milnor arcs $\gamma$ such that $p\in \overline {\gamma }$ and that $\lim f_{|\gamma }= c$.

Lemma 4.3 [Reference Durfee12, p. 1356]

(4.2)\begin{equation} {\rm{ind}}_{\infty}(f) = 1 + \sum_{p\in L^{\infty}, \ c\in {\mathbb{R}} \cup \{{\pm} \infty\}} i_{p,c}. \end{equation}

Proof. From definition 3.1, after identifying ${\mathbb {R}}^{2}$ with ${\mathbb {C}}$, we get the defining equality:

(4.3)\begin{equation} M(f) = \left\{ q\in {\mathbb{C}}\ \Big\vert\ \frac{\mathrm{{grad\ }} f(q)}{\|\mathrm{{grad\ }} f(q) \|} ={\pm} \frac{q}{\|q\|} \right\} . \end{equation}

Let us therefore consider the oriented $C^\infty$-map $\phi := \psi _{|C} \cdot \left (\frac {z}{\|z\|}\right )^{-1}: C \to S^1$, where $z$ denotes the variable in ${\mathbb {C}}$, and where both circles $C$ and $S^1$ are endowed with their counterclockwise orientation. The map $\phi$ is by definition the multiplication of $\psi _{|C}$ with the clockwise rotation $\left (\frac {z}{\|z\|}\right )^{-1} =\frac {\overline {z}}{\|z\|} : C \to S^{1}$ of degree $-1$. Using the winding number interpretation of the degree, one obtains the equality:

(4.4)\begin{equation} \deg (\phi) = \deg (\psi_{|C}) -1 = {\rm{ind}}_ \infty (f)-1. \end{equation}

Without loss of generality, we may assume that $1,-1 \in S^1$ are regular values of $\phi$. We obtain:

(4.5)\begin{equation} \deg (\phi) = \frac{1}{2} \left( \sum_{q \in \phi^{{-}1}(1)} {\rm{or}} (T_q \phi) + \sum_{q \in \phi^{{-}1}({-}1)} {\rm{or}} (T_q \phi) \right) = \sum_{q \in M(f) \cap C} \frac{1}{2} {\rm{or}} (T_q \phi), \end{equation}

where ${\rm {or}} (T_q \phi )$ denotes the orientation of the tangent map, and where the last equality follows since we have $\phi ^{-1}(\{-1, 1\}) = M(f)\cap C$ in view of (4.3).

Let us explain here what are the local orientations ${\rm {or}} (T_q \phi )$. Let $q \in \gamma \cap C$, for some Milnor arc $\gamma$. Referring to the definition in the beginning of §4.1, we have the following correspondences (where ‘increasing’ means here counterclockwise, and ‘decreasing’ means clockwise):

• • In the case (a) the Gauss map $\psi$ is increasing relative to the radial map $\frac {z}{\|z\|}$, and thus ${\rm {or}} (T_q\phi ) = +1$.

• • In the case (b) the Gauss map $\psi$ is decreasing relative to the radial map $\frac {z}{\|z\|}$, and therefore ${\rm {or}} (T_q\phi ) = -1$.

• • The case (c) at the point $q$ means that this point is a local maximum or a local minimum for the map $\phi$, thus a critical point. This situation cannot occur because we have assumed that $q$ is a regular point of $\phi$.

From (4.4) and (4.5) we then obtain:

\[ {\rm{ind}}_\infty (f) = 1 + \sum_{\gamma} i(\gamma), \]

where the sum runs over all the Milnor arcs of $f$.

In case $F_+\cup F_{-}$ contains a compact component, by lemma 2.2 we get that $|{\mathcal {L}}_{f}| =0$, and all fibres of $f$ are either compact and connected, or empty. A non-empty fibre of $f$ is then homomorphic to a circle. The winding number of $\mathrm {{grad\ }} f$ over such a circle is 1, and it follows that ${\rm {ind}}_{\infty }(f)=1$. Therefore, in the following we will tacitly consider only polynomials which have at least a non-compact fibre.

Let ${\rm {Sp}}(p, \lambda )$ and ${\rm {Va}}(p, \lambda )$ denote the numbers of connected components of fibres of $f$ outside the large disk $D$ which are splitting or vanishing, at the point $(p, \lambda )$, respectively.

Let ${\rm {Va}}(\pm \infty )$ denote the number of components of $F_+\cup F_{-}$. Note that there are two type of components which are counted in ${\rm {Va}}(\pm \infty )$: those which tend to a nontrivial segment of the line at infinity as the value of $f$ tends to infinity, and those which tend to a point $p\in L^{\infty }$ as the value of $f$ tends to infinity. These two types are illustrated in examples 2.7 and 6.3.

Theorem 4.4

(4.6)\begin{equation} {\rm{ind}}_{\infty}(f) = 1 + \frac12 \sum_{p\in {\mathcal{L}}_{f}, \lambda\in {\mathbb{R}}}{\rm{Sp}}(p, \lambda) - \frac12 \sum_{p\in {\mathcal{L}}_{f}, \lambda\in {\mathbb{R}}}{\rm{Va}}(p, \lambda) - \frac12 {\rm{Va}}({\pm} \infty) \end{equation}

Proof. By gathering the indices of the Milnor arcs of the same cluster, one recasts (4.2) as:

(4.7)\begin{equation} {\rm{ind}} _{\infty} (f) = 1 + \sum_{{\mathcal{C}}} \sum_{\gamma \in {\mathcal{C}}} i(\gamma), \end{equation}

where the sum runs over all Milnor clusters.

Let us compute the total index $\sum _{\gamma \in {{\mathcal {C}}}} i(\gamma )$ for each cluster ${\mathcal {C}}$. By corollary 4.2 the number of odd clusters of total index $+\frac {1}{2}$, or $-\frac {1}{2}$, is equal to the number ${\rm {Sp}}(p,\lambda )$, or ${\rm {Va}}(p,\lambda )$, respectively. Moreover, the number of clusters at $\pm \infty$ is equal to the number ${\rm {Va}}(\pm \infty )$ and the total index of each of these clusters is $-\frac {1}{2}$. Even clusters have total index $0$, thus do not contribute to the formula.

Our formula (4.6) follows by plugging in all these data in (4.7).

4.2. The local degree at infinity

We continue to consider a polynomial $f: {\mathbb {R}}^{2}\to {\mathbb {R}}$ of degree $d\ge 2$, and we prove here a key result that will be used for finding an upper bound of the index at infinity.

Remark 4.6 The inequality $d_{p}>0$ does not imply that $p\in {\mathcal {L}}_{f}$, like in the example $f= xy^{2}+x$, where $p:=[1:0:0]\in \{f_{d} =0\}$ with $d_{p}=2$, but $p\not \in {\mathcal {L}}_{f}$.

By this reason, out of the obvious inequalities:

(4.8)\begin{equation} d_{Re} \ge \sum_{p\in \{ f_{d} = 0\} \cap L^{\infty} } d_{p} \ge \sum_{p\in {\mathcal{L}}_{f}}d_{p} \end{equation}

the second may be strict, for instance in the example $f = x^{4}y + y^{3}$ where $d_{Re} =5$, but one has $\sum _{p\in {\mathcal {L}}_{f}}d_{p} =1$ because ${\mathcal {L}}_{f} = \{ [1:0:0]\}$.

Remark 4.7 Let $p\in \{f_{d}=0\}\cap L^{\infty }$. The following equality is displayed in [Reference Durfee12, lemma 7.3]:

(4.9)\begin{equation} {\rm{mult}}_{p}(\overline{M_{{\mathbb{C}}}(f)}, L^{\infty}_{{\mathbb{C}}}) =d_p-1. \end{equation}

We provide here an explicit proof of (4.9). One may assume (by an adequate linear change of coordinates) that $p= [1:0:0]$, and thus we have:

\[ f (x,y)= y^{d_{p}} r(x,y) + {\rm{l.o.t.}} \]

where $r$ is a homogeneous polynomial of degree $d-d_{p}$, and not divisible by $y$. In the chart $\{x \neq 0\}$, the Milnor set $\overline {M_{{\mathbb {C}}}(f)}$ has equation:

(4.10)\begin{equation} \hat{h}(y,z) ={-} d_p y^{d_p-1} r(1,y) - y^{d_p} r_{y}(1,y)+ y^{d_p+1} r_{x}(1,y) + z q(1,y,z) =0 \end{equation}

where $r_x$ and $r_y$ denote the partial derivatives of $r$, and $q(x,y,z)$ is a homogeneous polynomial of degree $d-1$. By our assumption, we also have $r(1,y)= c_0 + \cdots +c_ky^k$, where $c_0 \neq 0$, and $k\le d-d_p$. One has by definition:

\[ {\rm{mult}}_p(\overline{M_{{\mathbb{C}}}(f)}, L^\infty_{{\mathbb{C}}}) = {\rm{ord}}_{y}\left( \hat{h}(y,z)_{|L^\infty_{{\mathbb{C}}}} \right) \]

and due to (4.10), the later is precisely $d_p-1$.

5.1. Proof of theorem 5.1

Lemma 4.3 reads:

(5.5)\begin{equation} {\rm{ind}}_{\infty}(f) =1 + \sum_{p\in {\mathcal{L}}_{f }, c\in {\mathbb{R}}} i_{p,c} + \sum_{q\in L^{\infty}} i_{q,\infty}. \end{equation}

By (5.3), for each $p\in L^{\infty }\cap \{f_{d} =0\}$, in particular for $p\in {\mathcal {L}}_{f }$, we have the inequality $\sum _{c\in {\mathbb {R}}} i_{p,c} \le d_{p}-1$, for which we will evaluate the gaps studied in lemma 5.2.

To any complex non-real line $L\in {\rm {Cone}\ }\overline {M_{{\mathbb {C}}}(f)}_{p}$ there corresponds a positive number of complex non-real branches having $L$ as tangent at $p$. Each such complex non-real branch generates a gap of at least 1.

To any real line of ${\rm {Cone}\ }\overline {M_{{\mathbb {C}}}(f)}_{p}$ there may correspond real and complex non-real tangent branches. The non-real branches contribute, by lemma 5.2, with gaps of at least $1$.

A real tangent branch $\beta$ can be either:

1. (1) non-singular at $p$ and its two arcs are on both sides of $L^{\infty }$, or

2. (2) tangent to $L^{\infty }$ or singular at $p$, and thus yields a gap of at least 1, by lemma 5.2(c). According to lemma 5.2(c), this gap cumulates with the sign gaps of type (a).

We compute a lower bound for the total gap at $p$.

The ‘sign gaps,’ and the compensating exchange. The counterclockwise ordering of the Milnor arcs (cf. definition 3.5) induces an ordering in the subset of arcs at $p$. In turn, this induces an ordering among the real semi-lines of the real tangent cone ${\rm {Cone}\ } \overline {M(f)_{p}}$ which are on the same side of the line at infinity.

Let us assume that on the two sides of $L^{\infty }$ there are $r_{p}\ge 0$, respectively $s_{p}\ge 0$, semi-lines with tangent real Milnor arcs, and therefore $r_{p} + s_{p} \ge \deg (R_{p}^{{\mathrm {red}}})$. Applying proposition 3.16 to the consecutive arcs on each side we obtain that there are at least $\lfloor \frac {r_{p}}{2} \rfloor$ and $\lfloor \frac {s_{p}}{2} \rfloor$ arcs with non-positive index, respectively. According to lemma 5.2, each such arc produces a ‘sign gap’ of at least $\frac 12$. We thus obtain a total ‘sign gap’ of at least $\frac 12 \left (\lfloor \frac {r_{p}}{2}\rfloor + \lfloor \frac {s_{p}}{2} \rfloor \right )$, and notice that we have the inequality: $\lfloor \frac {r_{p}}{2} \rfloor + \lfloor \frac {s_{p}}{2} \rfloor \ge \left \lfloor \frac {\deg (R_{p}^{{\mathrm {red}}})-1}{2} \right \rfloor$.

Let us remark at this point that in the above computation, which starts with the ordered Milnor arcs at $p$, to be able to apply proposition 3.16 we ought to count all the Milnor arcs at $p$, thus not only the Milnor arcs for which $f$ tends to a finite value $c\in {\mathbb {R}}$ but also the Milnor arcsFootnote ⁶ for which $f$ tends to $\pm \infty$. And since the later Milnor arcs do not occur in the sum of (5.3), we should remove them from the above ‘sign gap’ count. Nevertheless, in the same time each of those contribute with a gap of type (b) of lemma 5.2, and this gap of $\frac 12$ is compensating the necessary removal of the corresponding ‘sign gap’ that has been counted as $\frac 12$ too. With this extra argument of compensating exchange, the above estimation of the ‘sign gap’ still holds.

The ‘singular gaps’ and the ‘complex gaps.’ The Milnor branches which are tangent at $p$ to the lines of the sub-variety $S_{p}$ correspond to case (c) of lemma 5.2 and each one produces a gap of at least 1 which adds up to the total sign gap. The same effect produces a complex non-real branch [by lemma 5.2(d)], while its tangent line at $p$ belongs to $K_{p}$. In both cases, the lines are considered with their multiple structure.

By adding up those gaps, we thus get the total gap at $p$ of at least $\frac 12 \left \lfloor \frac {\deg (R_{p}^{{\mathrm {red}}})-1}{2} \right \rfloor + \deg (S_{p})+ \deg (K_{p})$, i.e. the inequality:

(5.6)\begin{equation} \sum_{p\in {\mathcal{L}}_{f }, c\in {\mathbb{R}}} i_{p,c} \le d_{p}-1 - \frac12 \left\lfloor \frac{\deg(R_{p}^{{\mathrm{red}}})-1}{2} \right\rfloor - \deg(S_{p})- \deg(K_{p}) . \end{equation}

Finally we need to sum this up over all points $p\in {\mathcal {L}}_{f }$. Therefore we get from (4.8) the following inequality:

(5.7)\begin{equation} \sum_{p\in {\mathcal{L}}_{f }} d_{p}-1 \le d_{Re} - |{\mathcal{L}}_{f }|. \end{equation}

We have now to deal with the term $\sum _{q\in L^{\infty }} i_{q,\infty }$ in (5.5). By remark 3.8, to each fibre component counted by ${\rm {Va}}(\pm \infty )$ there corresponds injectively a cluster ${\mathcal {C}} = {\mathcal {C}}(\pm \infty )$, and for this cluster the sum of indices $\sum _{\gamma \in {\mathcal {C}}} i(\gamma )$ is $-1/2$ by corollary 4.2. By proposition 2.6, we have ${\rm {Va}}(\pm \infty ) \ge 2|{\mathcal {L}}_{f}|$. We thus obtain:

(5.8)\begin{equation} \sum_{q\in L^{\infty}} i_{q,\infty}= \sum_{{\mathcal{C}}}\sum_{\gamma \in {\mathcal{C}}} i(\gamma) \le -\frac12 2|{\mathcal{L}}_{f}| ={-}|{\mathcal{L}}_{f}| \end{equation}

where the first sum at the right hand side is taken over all clusters ${\mathcal {C}} ={\mathcal {C}}(\pm \infty )$.

Finally, plugging (5.6), (5.7) and (5.8) into (5.5), we obtain the claimed inequality (5.2).

We show how to compute an upper bound for the total gap in a different manner, by merging the set of singular Milnor branches (which projects onto the subset $S_{p}^{{\mathrm {red}}}$ of $R_{p}^{{\mathrm {red}}}$) into the total set of Milnor branches (which projects onto the set $R_{p}^{{\mathrm {red}}}$). This produces a more handy upper bound for the index at infinity, in particular formula (5.10) is in terms of the local degrees at infinity.

Corollary 5.4

(5.9)\begin{equation} {\rm{ind}}_{\infty}(f) \le 1 + d_{Re} - 2|{\mathcal{L}}_{f}| - \sum_{p\in L^{\infty}\cap \{f_{d} =0\}} \left(\left\lfloor \frac{\deg(R_{p}^{{\mathrm{red}}}) }{2} \right\rfloor + \deg(K_{p}) \right). \end{equation}

In particular$:$

(5.10)\begin{equation} {\rm{ind}}_{\infty}(f) \le 1 + d_{Re} - 2|{\mathcal{L}}_{f}| - \sum_{p\in L^{\infty}\cap \{f_{d} =0\}} \left\lfloor \frac{\delta_{p}}{2} \right\rfloor , \end{equation}

where $\delta _{p} := \deg {\rm {Cone}\ }\overline {M_{{\mathbb {C}}}(f)}_{p}^{{\mathrm {red}}_{{\mathbb {R}}}}$ is the degree of the cone in which the real line components are taken with reduced structure.

5.2. Revisiting Durfee's upper bound

Durfee [Reference Durfee12] has proved an upper bound (1.1) in terms of the degree $d$ only, namely ${\rm {ind}}_\infty f \le \max \{1, d-3\}$. Revisiting and completing Durfee's proof in [Reference Durfee12], we show here the following slight improvement:

Proof. (a). Follows immediately, either from (5.1) or from (5.2). (c). By lemma 2.2, the set $\{| f(x,y)| =R\}$ for $R\gg 1$ is diffeomorphic to a circle, and the winding number over a circle is $+1$, thus ${\rm {ind}}_{\infty }(f)=1$. This trivial fact was also observed in [Reference Durfee12, theorem 7.8].

(b). Let ${\mathcal {L}}_{f} = \{p\}$. Then, by a linear change of coordinates, we may assume that $p=[1:0:0]$. For $|{\mathcal {L}}_{f}| =1$, by theorem 4.4 and by Durfee's inequality (5.1), or by our improvement (5.10), we get:

(5.11)\begin{equation} {\rm{ind}}_{\infty}(f) = 1+ \frac12 \left( \sum_{ \lambda\in {\mathbb{R}}}{\rm{Sp}}(p, \lambda) - \sum_{\lambda\in {\mathbb{R}}\cup \{{\pm} \infty\}} {\rm{Va}}(p, \lambda) \right) \le d_{p} - 1 \end{equation}

If $d_{p} \le d-2$ then theorem 5.5 follows directly from (5.11). We consider in the following the two remaining cases: $d_{p} = d-1$ and $d_{p} = d$.

Case 1. $d_{p} = d-1$.

Proof. If $d_{p} = d-1 >0$ then one may assume, by an appropriate linear change of variables, that $p = [1:0:0]$ and $f_{d}=xy^{d-1}$. One therefore has$:$

\[ f(x,y)= xy^{d-1}+ xh(x,y) + u(y) \]

with $\deg h(x,y) < d-2$ and $\deg u(y)\le d-1$.

We will show that the projective closure $\overline {f^{-1}(0)}$ contains the point $[0:1:0] \in L^{\infty }$, which is different from $p$. We may assume that $u\not \equiv 0$, since if not, then $\{x=0\} \subset f^{-1}(0)$, thus $[0:1:0]\in {\mathcal {L}}_{f}$ and the claim is proved. Let then $b\in {\mathbb {R}}^{*}$ be the leading coefficient of $u$.

For every fixed $x_0 \in {\mathbb {R}}^{*}$, $x_{0} \not = -b$, the sign of the polynomial $f(x_0,y)$ of variable $y$, for $y\gg 1$, is the sign of its leading coefficient; this is $x_0+b$ in case $\deg u=d-1$, and it is $x_0$ in case $\deg u < d-1$. Let us choose $x_0 \in {\mathbb {R}}^*$ such that $b (x_0 +b)<0$, and thus $b x_0<0$ too. Then $f(x_0,y)f(0,y)<0$ for any $y\gg 1$. This implies that for any $y \gg 1$ there exists some value $t_{y}$ bounded between $0$ and $x_{0}$ such that $(t_{y}, y)\in f^{-1}(0)$. By taking the limit $y\to \infty$, this shows that $[0:1:0]\in \overline {f^{-1}(0)} \cap L^{\infty }$.

Lemma 5.7 shows that the case $d= d_{p} - 1$, with $p\in {\mathcal {L}}_{f}$ and $|{\mathcal {L}}_{f}|=1$ is impossible, confirming Durfee's claim.

Case 2. $d_{p} = d$.

We may assume as above that $p=[1:0:0]$ and since $d_{p} = d$, one may also assume that, modulo some appropriate linear change of coordinates, one has $f_{d} =y^{d}$.

Proof. Let us assume that $f$ contains mixed terms, namely let $f = y^{d}+ xyq(x,y) +v(x) + u(y)$, where $q \not \equiv 0$ is a polynomial of degree $\le d-3$, and $v(x)$ and $u(y)$ are some polynomials of degrees $\le d-1$. We write explicitly the equation $\hat h(1,y,z) =0$ of the closure $\overline {M(f)}$ of the Milnor set in the chart $\{x=1\}$. This is a polynomial of degree at least $d-1$ because it contains the term $dy^{d-1}$. Its order at $(0,0)$ is ${\rm {ord}} \hat h(1,y,z) \le {\rm {ord}} z\hat q(1,y,z) + zy\hat q(1,y,z) \le d-2$, where $\hat q(x,y,z)$ denotes the homogenization of $q(x,y)$ of degree $d-3$ by the variable $z$. Thus all the terms of $\hat h(1,y,z)$ of degree $< d-1$ contain $z$. This implies that $\overline {M(f)}_{p}$ contains a (complex) branch which is tangent to the line at infinity $L^{\infty } = \{z=0\}$.

Let us now treat the complementary case, i.e. whenever $f$ contains no mixed terms, i.e. $f= y^{d} +v(x) + u(y)$, where $v(x)$ and $u(y)$ are some polynomials of degrees $\le d-1$. Then the Milnor set germ $\overline {M(f)}_{p}$ in the chart $\{x =1\}$ is defined by the equation:

(5.12)\begin{equation} \hat{h}(1,y,z)={-} d y^{d-1} + zy\hat{v}_x(1,z) - z \hat{u}_y(y,z), \end{equation}

where $\hat {v}_x(x,z)$ and $\hat {u}_y(y,z)$ denote the homogenization of degree $d-2$ of the derivatives $v_x$ and $u_y$. The 1st and the 3rd terms of (5.12) are homogeneous of degree $d-1$, while the term in the middle is of order $\le d-2$ if and only if $\deg v \ge 3$. We deduce that the tangent cone ${\rm {Cone}\ } _{p}\overline {M(f)}$ contains the line $\{z=0\}$ if and only if $\deg v \ge 3$.

We compute the index in the special case of lemma 5.8.

We continue the proof of Case 2. By lemma 5.8 and by lemma 5.9, if there is no Milnor branch tangent to $L^{\infty }$ at $p$, then we obtain ${\rm {ind}}_{\infty }(f) =0$ when $d =3$, and ${\rm {ind}}_{\infty }(f) \le d-3$ when $d>3$, hence theorem 5.5 is proved in this situation.

In what follows we focus on the last remaining case established by lemma 5.8, namely: there exists at least one (complex) Milnor branch $\beta$ tangent to $L^{\infty }$ at $p$. The study falls into the following 4 situations:

1. (i) There are at least two Milnor branches at $p$ which are tangent to $L^{\infty }$, then by lemma 5.2(c) we get a gap of at least 1 for each of these branches. Therefore ${\rm {ind}}_{\infty }(f) \le d_{p}-1-2= d_{p} - 3$.

2. (ii) There is a single Milnor branch $\beta$ tangent to $L^{\infty }$ at $p$, and such that ${\rm {mult}}_{p}(\beta, L^{\infty }_{{\mathbb {C}}}) \ge 2$. By (4.9), this implies $d_{p}\ge 3$.

   If ${\rm {mult}}_{p}(\beta, L^{\infty }_{{\mathbb {C}}}) >2$ then we have a gap of at least 2, and thus ${\rm {ind}}_{\infty }(f) \le d_{p}-3$. If ${\rm {mult}}_{p}(\beta, L^{\infty }_{{\mathbb {C}}}) = 2$ and the indices of the arcs of $\beta$ are not both $+\frac 12$, then we get a gap of at least $3/2$. Since the index is an integer, the gap is of at least 2, and therefore ${\rm {ind}}_{\infty }(f) \le d_{p}-3$ again.

   Last case, let ${\rm {mult}}_{p}(\beta, L^{\infty }_{{\mathbb {C}}}) = 2$ and such that both arcs of $\beta$ have index $+\frac 12$. Since $\beta$ is also tangent to $L^{\infty }$, it follows that $\beta$ is nonsingular at $p$, more precisely the germ of $\beta$ at $p$ is equivalent, after some linear change of coordinates, with the curve $z=y^{2}$. Thus the two Milnor arcs are in the same half-plane of the chart ${\mathbb {R}}^{2}$ cut by the line $z=0$. According to proposition 3.15, after each of the two splittings we obtain two components of the fibres of $f$ which are tangent to the line $L := \{z=0\}$, and moreover, along the two Milnor arcs the tangency is to different semi-lines, say $L_+$ and $L_{-}$. This implies that, in the absence of other splittings or vanishings at $p$, there should exist a trivial fibration connecting two components tangent to different semi-lines, which is treated by proposition 3.16, and which tells that this situation is impossible.

3. (iii) There is a single Milnor branch tangent to $L^{\infty }$, and there are also non-tangent Milnor branches at $p$, such that at least one of the Milnor arcs has index $-\frac 12$. Then by lemma 5.2(a) and (c) we obtain an index gap of at least 2. Therefore we get ${\rm {ind}}_{\infty }(f) \le d_{p}-3$.

4. (iv) There is a single tangent Milnor branch at $p$, there are one or more transversal Milnor branches, and such that all the Milnor arcs at $p$ have index $+\frac 12$. This means that all arcs are of splitting type, and in particular each arc is a cluster. Our proposition 3.16 shows that this situation is impossible.