Mathematische Annalen

, Volume 361, Issue 3–4, pp 1049–1054

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

Open Access
Article

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.

Keywords

Singular Point General Fiber Bifurcation Locus Cartier Divisor Newton Polyhedron
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 ), 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 : $$f(x,y) = x + x^2y$$, 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  in terms of Newton polyhedra at infinity. An estimation in terms of the degree $$d$$ of $$f$$ was given by Gwoździewicz and Płoski : 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  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  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:
1. (a)

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

2. (b)

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

3. (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:
1. (d)

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

2. (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 . Let us recall it together with some properties which we shall use.

Definition 2.1

 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:
1. (a)

$$X$$ has at most cyclic quotient singularities.

2. (b)

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

3. (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  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. $$\square$$

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