Bifurcation locus and branches at infinity of a polynomial $f:\mathbb {C}^2\rightarrow \mathbb {C}$

Abstract

We show that the number of bifurcation values at infinity of a polynomial function $f:\mathbb {C}^2\rightarrow \mathbb {C}$ is at most the number of branches at infinity of a general fiber of $f$ and that this upper bound can be diminished by one in certain cases.

1 Introduction

Let $f:\mathbb {C}^2 \rightarrow \mathbb {C}$ be a polynomial function in a fixed coordinate system. It is well known (as being proved originally by Thom [17]), that $f$ is a locally trivial $C^\infty$ fibration outside a finite subset of the target. The smallest such set is called the bifurcation set of $f$ and will be denoted here by $B(f)$. The set $B(f)$ might be larger than the set of critical values $f(\mathrm{{Sing}}f)$, like for instance in the following simple example due to Broughton [1]: $f(x,y) = x + x^2y$, where $\mathrm{{Sing}}f = \emptyset$ but $B(f) = \{ 0\}$, and we say that $0$ is a critical value at infinity of $f$. The set $B_\infty (f)$ of bifurcation values at infinity, or critical values at infinity, consists of points $a\in \mathbb {C}$ at which the restriction of $f$ to the complement of a large enough ball (centred at $0\in \mathbb {C}^2$) is not a locally trivial bundle. There are several criteria to detect such a value; one may consult e.g. [2, 3, 5, 16, 18, 19]. For instance: $a\in B_\infty (f)$ if and only if there exists a sequence of points $(p_k)_{k\in \mathbb {N}}\subset \mathbb {C}^2$ such that $\Vert p_k\Vert \rightarrow \infty , \mathrm{grad}\ f(p_k)\rightarrow 0$ and $f(p_k)\rightarrow a$ as $k\rightarrow \infty$.

Upper bounds for $\# B_\infty (f)$ have been found in the 1990’s by Lê and Oka [12] in terms of Newton polyhedra at infinity. An estimation in terms of the degree $d$ of $f$ was given by Gwoździewicz and Płoski [8]: if $\dim \mathrm{{Sing}}f \le 0$ then $\# B_\infty (f)\le \max \{ 1, d-3\}$. In the general case (dropping the condition $\dim \mathrm{{Sing}}f \le 0$) we have $\# B_\infty (f)\le d-1$, see e.g. [10, 11]. Recently Gwoździewicz [9] proved the following estimation of $\# B_\infty (f)$: if $\nu _0$ denotes the number of branches at infinity of the (reduced) fibre $f^{-1}(0)$, then the number of critical values at infinity other than 0 is at most $\nu _0$. Here we refine and improve this statement by using a different method, in which results by Miyanishi [13, 14] and Gurjar [6] play an important role.

For $a\in \mathbb {C}$, let us denote by $\nu _a$ the number of branches at infinity of the reduced fiber $f^{-1}(a)$. This number is equal to $\nu _\mathrm {gen}$ for all values $a\in \mathbb {C}$ except finitely many for which one may have either $\nu _a<\nu _\mathrm {gen}$ or $\nu _a>\nu _\mathrm {gen}$. Let $\nu _{\mathrm{{min}}} := \mathrm{{inf}}\{ \nu _a \mid a\in \mathbb {C}\}$. Let us denote by $b$ the number of points at infinity of $f$, i.e. $b := \# \overline{f^{-1}(a)} \cap L_\infty$, where $L_\infty$ is the line at infinity $\mathbb {P}^2{\setminus }C^2$.

Under these notations, our main result is the following:

Theorem 1.1

Let $f:\mathbb {C}^2\rightarrow \mathbb {C}$ be a polynomial function of degree $d$. Then:

• (a) $\# B_\infty (f) \le \mathrm{{min}}\{\nu _\mathrm {gen},\nu _{\mathrm{{min}}} + 1 \}$.

• (b) $\# \{ a\in \mathbb {C}\mid \nu _a< \nu _\mathrm {gen}\} \le \nu _\mathrm {gen}- b$.

• (c) $\# \{ a\in \mathbb {C}\mid \nu _a> \nu _\mathrm {gen}\} \le \nu _{\mathrm{{min}}}$ (this remains true even if we count branches with multiplicities).

In case $\nu _\mathrm {gen}> \frac{d}{2}$, we moreover have:

• (d) $\# B_\infty (f) \le \mathrm{{min}}\{\nu _\mathrm {gen}-1,\nu _{\mathrm{{min}}} \}$.

• (e) $\# \{ a\in \mathbb {C}\mid \nu _a> \nu _\mathrm {gen}\} \le \nu _{\mathrm{{min}}}-1$ (this remains true even if we count branches with multiplicities).

Remark 1.2

Point (a) of Theorem 1.1 is equivalent to Gwoździewicz’s [9, Theorem 2.1]. His result is a by-product of the local study of pencils of curves of Yomdin-Ephraim type. Our method is totally different and allows us to prove moreover several new issues, namely (b)–(e) of Theorem 1.1.

Remark 1.3

As Gwoździewicz remarks, his inequality [9, Theorem 2.1] is “almost” sharp, i.e. not sharp by one. Our new inequality (d) improves by one the inequality (a) under the additional condition $\nu _\mathrm {gen}> \frac{d}{2}$, thus yields the sharp upper bound, as shown by the example $f:\mathbb {C}^2 \rightarrow \mathbb {C}$, $f(x,y) = x + x^2 y$, where $d = \deg f = 3$, $\nu _{\mathrm{{min}}} =\nu _\mathrm {gen}= 2$, $b=2$ and $B_\infty (f) = \{ 0\}$ with $\nu _0 = 3$.

The same example shows that our estimations (b) and (e) are also sharp.

2 Proof of Theorem 1.1

We need here the important concept of affine surfaces which contain a cylinder-like open subset which was introduced by Miyanishi [13]. Let us recall it together with some properties which we shall use.

Definition 2.1

[14] Let $X$ be a normal affine surface. We say that $X$ contains a cylinder-like open subset $U$, if there exists a smooth curve $C$ such that $U\cong \mathbb {C}\times C$.

Let $X$ be as in the above definition and let $\pi : U\rightarrow C$ be the projection. After [14, p.194], the projection $\pi$ has a unique extension to a $\mathbb {C}$-fibration $\rho : X\rightarrow \bar{C}$, where $\bar{C}$ denotes the smooth completion of the curve $C$. We have the following important result of Gurjar and Miyanishi:

Theorem 2.2

[6, 7, 13] Let $X$ be a normal affine surface with a $\mathbb {C}$-fibration $f : X\rightarrow B$, where $B$ is a smooth curve. Then:

• (a) $X$ has at most cyclic quotient singularities.

• (b) Every fiber of $f$ is a disjoint union of curves isomorphic to $\mathbb {C}$.

• (c) A component of a fiber of $f$ contains at most one singular point of $X$. If a component of a fiber occurs with multiplicity $1$ in the scheme-theoretic fiber, then no singular point of $X$ lies on this component.$\square$

Corollary 2.3

Let $X$ be a normal affine surface, which contains a cylinder-like open subset $U$. Then the set $X{\setminus } U$ is a disjoint union of curves isomorphic to $\mathbb {C}$. Moreover, every connected component $l_i$ of this set contains at most one singular point of $X$. $\square$

Let $f:\mathbb {C}^2\rightarrow \mathbb {C}$ be a polynomial function in fixed affine coordinates and denote by $\tilde{f}(x,y,z)$ the homogenization of $f$ by a new variable $z$, namely $\tilde{f}(x,y,z)=f_d+ zf_{d-1}+\cdots +z^d f_0$. Let $X := \{ ([x:y:z],t)\in \mathbb {P}^2\times \mathbb {C}\mid \tilde{f}(x,y,z) = tz^d\}$ be the closure in $\mathbb {P}^2 \times \mathbb {C}$ of the graph $\Gamma := \mathrm{{graph}}(f)\subset \mathbb {C}^2\times \mathbb {C}$. Then $X$ is a hypersurface and the points at infinity of $X$ (i.e. points outside of $\Gamma$) forms precisely the set $\{a_1,\ldots , a_b\} \times \mathbb {C}$, where $\{a_1,\ldots ,a_b\}$ are all points at infinity of the curve $f=0$. In particular if $\rho : \mathbb P^2\times \mathbb {C}\rightarrow \mathbb P^2$ denotes the first projection, then $\rho (X{\setminus } \Gamma )=\{a_1,\ldots ,a_b\}.$

The second projection $\pi : X \rightarrow \mathbb {C}$, $(x,t)\mapsto t$, is a proper extension of $f$. Let $\nu : X'\rightarrow X$ be the normalization of $X$. Composing $\nu$ with $\pi$ yields $\pi ': X'\rightarrow \mathbb {C}$, which is also a proper extension of $f$. We shall denote it by $\tilde{f}$ in the following.

On the other side composing $\nu$ with $\rho$ yields $\rho ': X'\rightarrow \mathbb P^2$ and $\rho '(X'{\setminus } \Gamma )=\{a_1,\ldots ,a_b\}$, i.e., the points at infinity of $X'$ lie over the points $\{a_1,\ldots ,a_b\}.$

Lemma 2.4

The set $X'{\setminus } \Gamma$ is a disjoint union of affine curves, $l_1,\ldots ,l_r$, each curve $l_i$ is isomorphic to $\mathbb {C}$. On each line $l_i$ there is at most one singular point of $X'$. Moreover, $b\le r\le \nu _{\mathrm{{min}}}$.

Proof

Let us choose a line $l\subset \mathbb P^2$ such that $l\cap \{ a_1,\ldots ,a_b\}=\emptyset$. Let $X_1:=(\mathbb P^2{\setminus } l)\times \mathbb {C}\cap X$. The surface $X_1$ is affine and $X_1'{\setminus } \Gamma =\bigcup ^r_{i=1} l_i$, where $X_1'$ denotes the normalization of $X_1$. The surfaces $X'$ and $X_1'$ have the same points at infinity since there is no points at infinity of $X'$ which belongs to the line $l$.

Since the surface $X_1'$ contains a cylinder-like open subset $U := graph(f_{|\mathbb {C}^2{\setminus } l})\cong \mathbb {C}\times \mathbb {C}^*$ and $X_1'{\setminus } U=\bigcup ^r_{i=1} l_i$, the first part of our claim follows from Corollary 2.3. Next, the map $\tilde{f}$ restricted to $l_i$ is finite, hence surjective. This implies that every fiber of $\tilde{f}$ has a branch at infinity which intersects $l_i$. In particular $r\le \nu _{\mathrm{{min}}}$. The inequality $r\ge b$ is obvious. $\square$

Denote by $f_i : l_i\cong \mathbb {C}\rightarrow \mathbb {C}$ the restriction of $\tilde{f}$ to $l_i$. It can be identified with a one variable polynomial, the degree of which is equal to the number $\nu _i$ of branches of a generic fiber of $\tilde{f}$ which intersect $l_i$. In particular $\sum ^r_{i=1} \nu _i=\nu _\mathrm {gen}$.

The polynomial $f_i$ of degree $\nu _i$ can have at most $\nu _i-1$ critical points. If a fiber $\tilde{f}^{-1}(a)$ does not contain critical points of any $f_i$ and does not contain singular points of $X'$, then the point $a\not \in B_\infty (f)$. This follows from general arguments concerning Whitney stratifications and Thom Isotopy Lemma, like in [3, 15, 19], but let us outline a short proof here. Firstly, the fiber $\tilde{f}^{-1}(a)$ cannot contain multiple components since otherwise, for some $i$, the fiber $f_i^{-1}(a)$ will also have a multiple component, thus a singularity, which contradicts our assumption. Therefore the fiber $\tilde{f}^{-1}(a)$ is nonsingular outside some large ball $B(0,R)\subset \mathbb {C}^2$. By the Sard Theorem there is a real value $R'>R$ such that the sphere $\partial B(0,R')$ is transversal to $\tilde{f}^{-1}(a)$. In particular there is a small disc $U(a,\rho )$ such that for every $b\in U(a,\rho )$ the fiber $\tilde{f}^{-1}(b)$ is smooth outside $B(0,R)$ and it is transversal to $\partial B(0,R').$ We can also assume that $\rho$ is so small that $\tilde{f}^{-1}(b)$ does not contain critical points of any of the polynomials $f_i$, for $i=1,\ldots ,r$, and it does not contain any singular point of $X'.$ This means in particular that all these fibers are transversal to all curves $l_i, \ i=1,\ldots , r.$ Now take $Y=\tilde{f}^{-1}(U(a,\rho )){\setminus } Int(B(0,R').$ It is a smooth manifold with boundary, where the boundary $\partial Y$ is $\partial B(0,R')\cap \tilde{f}^{-1}(U(a,\rho ))$. The set $V :=(\bigcup ^r_{i=1} l_i)\cap Y$ is a smooth submanifold of $Y$. The mapping $g:=\tilde{f}_{|Y} : Y \rightarrow U(a,\rho )$ is proper and all fibers of $g$ are transversal to $V$ and to $\partial Y.$ By the Ehresmann Theorem [4] there is a trivialization of $g$ which preserves $V$ and $\partial Y.$ This proves our claim that $a\not \in B_\infty (f)$.

Finally we conclude that the bifurcation values at infinity for $f$ can be only images by $\tilde{f}$ of critical points of $f_i, \ i=1,\ldots , r$ and images of singular point of $X'.$ Summing up, we get that $f$ can have at most $\nu _\mathrm {gen}$ critical values at infinity, which shows one of the inequalities of point (a). Moreover, the inequality $\nu _a<\nu _\mathrm {gen}$ is possible only if $a$ is a critical value of some polynomial $f_i$. This means that $\# \{ a\in \mathbb {C}\mid \nu _a< \nu _\mathrm {gen}\} \le \sum ^r_{i=1} (\nu _i-1)\le \nu _\mathrm {gen}- r\le \nu _\mathrm {gen}-b$, which proves (b).

Let us assume now $\nu _a=\nu _{\mathrm{{min}}}$. We have $\nu _a \ge \sum ^r_{i=1}\#\{ x\in l_i \mid f_i(x)=a\}$ since in every such point $x$ there is at least one branch at infinity of the fiber $f^{-1}(a)$. Note that if $f_i(x)=a$ then $\mathrm{{ord}}_x (f_i-a) =\mathrm{{ord}}_x f_i'+1$. Thus:

$$\begin{aligned} \#\{ x\in l_i \mid f_i(x)=a\} = \sum _{ x\in l_i, f_i(x)=a } [\mathrm{{ord}}_x (f_i-a)- \mathrm{{ord}}_x f_i']. \end{aligned}$$

We have clearly the equality $\sum _{x\in l_i} \mathrm{{ord}}_x (f_i-a)=\nu _i$. Hence

$$\begin{aligned} \sum _{ x\in l_i, f_i(x)=a } [\mathrm{{ord}}_x (f_i-a)- \mathrm{{ord}}_x f_i']= \nu _i - \sum _{ x\in l_i, f_i(x)=a } \mathrm{{ord}}_x f_i'. \end{aligned}$$

Since $\sum _{ x\in l_i} \mathrm{{ord}}_x f_i'=\nu _i-1$ we have:

$$\begin{aligned} \nu _i - \sum _{ x\in l_i, f_i(x)=a } \mathrm{{ord}}_x f_i' = 1 + \sum _{x\in l_i, f_i(x)\not =a} \mathrm{{ord}}_x f_i'. \end{aligned}$$

Note that:

$$\begin{aligned} 1\!+\! \sum _{x\in l_i, f_i(x)\not =a} \mathrm{{ord}}_x f_i'\ge \#\{ x\in l_i \mid f(x)\not =a, \text{ and } \text{ either } f'_i (x)\!=\!0\ \text{ or } \ x \in \mathrm{{Sing}}(X')\}. \end{aligned}$$

The number at the right side is greater or equal to the number of critical values at infinity of $f$ different from $a$. Finally, taking the sum over all $i\in \{1,\ldots , r\}$ we get $\# B_\infty (f) \le \nu _{\mathrm{{min}}}+1$, which completes the proof of (a).

To prove (c), note that if the fiber $\tilde{f}^{-1}(a)$ does not contain a singular point of $X'$, which lies on some $l_i$, then the intersection multiplicity $\overline{l_i}\cdot \tilde{f}^{-1}(a)$ is equal to $\nu _i= \deg f_i$, where we consider here $\tilde{f}^{-1}(a)$ as a scheme-theoretic fiber of $\tilde{f}$. Hence the fiber $\tilde{f}^{-1}(a)$ has at most $\nu _i$ branches on $l_i$ (even if counted with multiplicity). This implies $\nu _a\le \nu _\mathrm {gen}$. Therefore $\# \{ a\in \mathbb {C}\mid \nu _a> \nu _\mathrm {gen}\} \le r\le \nu _{\mathrm{{min}}}$.

To prove (d) and (e) it is enough to show that if $\nu _\mathrm {gen}> \frac{d}{2}$, then at least one line $l_i$ does not contain singular points of $X'$. Let $d_i$ be the smallest positive integer such that $d_il_i$ is a Cartier divisor in $X'$ (such a number exists because $X'$ has only cyclic singularities). Since $l_i$ is smooth, we have that $d_i=1$ if and only if the line $l_i$ does not contain any singular point of $X'$, by the following lemma, the proof of which is left to the reader:

Lemma 2.5

Let $X^n$ be an algebraic variety and let $Z^r\subset X^n$ be a subvariety which is a complete intersection in $X^n$. If a point $z\in Z^r$ is nonsingular on $Z^r$, then it is nonsingular on $X^n$. $\square$

Now let $Z$ be the closure of $\Gamma$ in $\mathbb P^2\times \mathbb P^1$ and let $Z'$ denote its normalization. We have clearly the inclusion $X'\subset Z'$. Let $\Pi : Z'\rightarrow \mathbb P^2$ the first projection, where the second projection $Z'\rightarrow \mathbb P^1$ is an extension of $\tilde{f}$ which we will denote by $\tilde{f}'$. Note that for $a\not =\infty$ fibers $\tilde{f}^{-1}(a)$ and $(\tilde{f}')^{-1}(a)$ coincide.

Let $(\tilde{f}')^{-1}(\infty )=S_1\cup \cdots \cup S_k$ (where $S_i$ are irreducible and taken with reduced structure). Recall that $L_\infty =\mathbb P^2{\setminus } \mathbb {C}^2$ is the line at infinity. We have $\Pi ^*(L_\infty )=\sum _{i=1}^k m_iS_i+\sum ^r_{i=1} e_i\overline{l_i}$. Since $\Pi ^*(L_\infty )$ is a Cartier divisor we have $e_i=n_id_i,$ where $n_i$ is a positive integer.

Let us assume that every line $l_i$ contains some singular point of $X'$, i.e., that $d_i> 1$ for any $i$. Denoting by $F\subset \mathbb P^2$ the closure of a general fiber of $f$, since $\Pi$ is a birational morphism, we have:

$$\begin{aligned} d= F\cdot L_\infty =\Pi ^*(F)\cdot \Pi ^*(L_\infty )=\left( \tilde{f}'\right) ^*(a)\cdot \left( \sum _{i=1}^k m_iS_i+\sum ^r_{i=1} e_i\overline{l_i}\right) . \end{aligned}$$

Note that $\Pi ^*(F)\cdot \sum ^k_{i=1} m_iS_i=0$ since $|(\tilde{f}')^*(a)|\cap |\sum ^k_{i=1} m_iS_i|=|(\tilde{f}')^*(a)|\cap |(\tilde{f}')^*(\infty )|=\emptyset$. Moreover we have $\nu _i=(\tilde{f}')^*(a) \cdot \overline{l_i}$. Thus:

$$\begin{aligned} d=\sum ^r_{i=1} n_id_i\nu _i\ge \sum ^r_{i=1} 2\nu _i=2\nu _\mathrm {gen}\end{aligned}$$

and this ends our proof. $\square$