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

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## 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## 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**

- (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).

- (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**

- (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).

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.

## Notes

### Acknowledgments

The authors are grateful to Professor R.V. Gurjar from Tata Institute and Professor K. Palka from IMPAN for helpful discussions. They are also thank the referee for helpful comments.

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