Bifurcation locus and branches at infinity of a polynomial f : C2 ->c

We show that the number of bifurcation points at infinity of a polynomial function f : C2 ->C is at most the number of branches at infinity of a generic fiber of f and that this upper bound can be diminished by one in certain cases.


Introduction
Let f : C 2 → C be a polynomial function in a fixed coordinate system. It is well known (as being proved originally by Thom [T]), that f is a locally trivial C ∞ 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 f (Singf ) of critical values of f since it contains also the set B ∞ (f ) of bifurcation points at infinity. Roughly speaking, B ∞ (f ) consists of points at which the restriction of f to a neighbourhood of infinity (i.e. outside a large enough ball) is not a locally trivial bundle. We say that a ∈ B ∞ (f ) is a critical value at infinity of f . There are several criteria to detect such a value, one may consult e.g. [Su], [HL], [Ti1], [Du], [CK], [Ti2]. For instance, a ∈ B ∞ (f ) if and only if there exists a sequence of points (p k ) k∈N ⊂ C 2 such that p k → ∞, grad f (p k ) → 0 and f (p k ) → a as k → ∞.
Upper bounds for #B ∞ (f ) have been found in the 1990's by Lê V.T. and M. Oka [LO] in terms of Newton polyhedra at infinity. An estimation in terms of the degree d of f was given by Gwoździewicz and Płoski [GP] In the general case (dropping the condition dim Singf ≤ 0) we have #B ∞ (f ) ≤ d − 1, see e.g. [Je], [JK]. Recently Gwoździewicz [Gw] proved the following estimation of #B ∞ (f ): if ν 0 denotes the number of branches at infinity of the fibre f −1 (0), then the number of critical values at infinity other than 0 is at most ν 0 . Here we refine and improve this statement by using a different method, in which results by Miyanishi [Mi1], [Mi] and Gurjar [Gu] play an important role.
For a ∈ C, let us denote by ν a the number of branches at infinity of the fiber f −1 (a). This number is equal to ν gen for all values a ∈ C except finitely many for which one may have either ν a < ν gen or ν a > ν gen . Let ν min := inf{ν a | a ∈ C}. Let us denote by b the number of points at infinity of f , i.e. b := #f −1 (a) ∩ L ∞ , where L ∞ is the line at infinity P n \ C n .
Under these notations, our main result is the following: Theorem 1.1. Let f : C 2 → C be a polynomial function of degree d. Then: Date: January 28, 2014. The first author was partially supported by Université Lille 1 and by the grant of NCN, 2014-2017.
In case ν gen > d 2 , we moreover have: Remark 1.2. Point (a) of Theorem 1.1 is equivalent to Gwoździewicz's [Gw,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 [Gw, 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 ν gen > d 2 , thus yields the sharp upper bound, as shown by the example f : The same example shows that our estimations (b) and (e) are also sharp.
Acknowledgements. The authors are grateful to Professor R.V. Gurjar from Tata Institute and Professor K. Palka from IMPAN for helpful discussions.

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 M. Miyanishi [Mi1]. Let us recall it together with some properties which we shall use.
Definition 2.1. [Mi] Let X be a normal affine surface. We say that X contains a cylinderlike open subset U , if there exists a smooth curve C such that U ∼ = C × C.
Let X be as in the above definition and let π : U → C be the projection. After [Mi,p.194], the projection π has a unique extension to a C-fibration ρ : X →C, whereC denotes the smooth completion of the curve C. We have the following important result of Gurjar and Miyanishi: Let f : C 2 → C be a polynomial function in fixed affine coordinates and denote bỹ f (x, y, z) the homogenization of f by a new variable z, namelyf (x, y, z) = f d + zf d−1 + ... + z d f 0 . Let X := {([x : y : z], t) ∈ P 2 × C |f (x, y, z) = tz d } be the closure in P 2 × C of the graph Γ := graph(f ) ⊂ C 2 × C. Then X is a hypersurface and the points at infinity of f are precisely the set {a 1 , ..., a n } : Let X ′ be the normalization of X. Composing the normalisation map with π yields π ′ : X ′ → C, which is also a proper extension of f . For simplicity we shall denote it again by f in the following.
Lemma 2.4. The set X ′ \ Γ is a disjoint union of affine lines, l 1 , ..., l r , each line l i is isomorphic to C. On each line l i there is at most one singular point of X ′ . Moreover, b ≤ r ≤ ν min .
Proof. Let us choose a line l ⊂ P 2 such that l ∩ {a 1 , ..., a n } = ∅. Let X 1 := (P 2 \ l) × C ∩ X ′ . The surface X 1 is affine and X ′ 1 \ Γ = r i=1 l i , where X ′ 1 denotes the normalization of X 1 . Since the surface X ′ 1 contains a cylinder-like open subset U ∼ = C 2 \l = C×C * , the first part of our claim follows from Corollary 2.3. Next, the map f restricted to l i is finite, hence surjective. This implies that every fiber of f has a branch at infinity which intersects l i . In particular r ≤ ν min . The inequality r ≥ b is obvious.
Denote by f i : l i ∼ = C → C the restriction of f to l i . It can be identified with a one variable polynomial, the degree of which is equal to the number ν i of branches of a generic fiber of f which intersect l i . In particular r i=1 ν i = ν gen . The polynomial f i of degree ν i can have at most ν i − 1 singular points and their images by f i are bifurcation values at infinity for f (this can be easily deduced from [Eh]). There might also be at most one singular point of X ′ which lies on l i , and then its image by f is possibly a bifurcation value too. Summing up, we get that f can have at most ν gen critical values at infinity, which shows one of the inequalities of point (a). Moreover, the inequality ν a < ν gen is possible only if a is a critical value of some polynomial f i . This means that #{a ∈ C | ν a < ν gen } ≤ r i=1 (ν i − 1) ≤ ν gen − r ≤ ν gen − b. This proves (b). Let us assume now ν a = ν min . We have ν a ≥ r i=1 #{x ∈ l i | 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 We have clearly the equality x∈l i ord x (f i − a) = ν i . Hence Note that: The number at the right side is greater or equal to the number of critical values at infinity of f different from f −1 (a). Finally, taking the sum over all i ∈ {1, . . . , r} we get #B ∞ (f ) ≤ ν min + 1, which completes the proof of (a).
To prove (c), note that if the fiber f −1 (a) does not contain singular points of X ′ , then the intersection multiplicity l i · f −1 (a) is equal to ν i = deg f i , hence the fiber f −1 (a) has at most ν i branches on l i . This implies ν a ≤ ν gen . Hence #{a ∈ C | ν a > ν gen } ≤ r ≤ ν min .
To prove (d) and (e) it is enough to show that if ν gen > 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 i l 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 ′ , see Lemma 2.5 below. Now let Z be the closure of Γ in P 2 × P 1 and let Z ′ denote its normalization. We have clearly the inclusion X ′ ⊂ Z ′ . Let Π : Z ′ → P 2 the first projection, where the second projection Z ′ → P 1 is an extension of f which we will denote for simplicity by f again. Let f −1 (∞) = S 1 ∪ ... ∪ S k (where S i are irreducible and taken with reduced structure). Recall that L ∞ = P 2 \ C 2 is the line at infinity. We have Let us assume that any line l i contains some singular point of X ′ , i.e., that d i > 1 for any i. Denoting by F ⊂ P 2 the closure of a general fiber of f = a, since Π is a birational morphism, we have: 2ν i = 2ν gen and this ends our proof. ✷ Lemma 2.5. Let X n be an algebraic variety and let Z r ⊂ X n be a subvariety which is a complete intersection in X n . If a point z ∈ Z r is nonsingular on Z r , then it is nonsingular on X n .
Proof. Let us recall that if X n is an algebraic variety, then a point z ∈ X n is smooth, if and only if the local ring O z (X) is regular. This is equivalent to the fact that dim C m/m 2 = dim X n = n, where m denotes the maximal ideal of O z (X n ).
Let I(Z r ) = {f 1 , ..., f l } ⊂ C[X n ], where l = n − r, be the ideal of Z r in the ring C[X n ]. We have O z (Z r ) = O z (X n )/(f 1 , ..., f l ). In particular if m ′ denotes the maximal ideal of O z (Z r ) and m denotes the maximal ideal of O z (X n ) then m ′ = m/(f 1 , ..., f l ). Let α i denote a class of the polynomial f i in m/m 2 . Let us note that (2.1) m ′ /m ′ 2 = m/(m 2 + (α 1 , ..., α l )). Since the point z is smooth on Z r we have dim C m ′ /m ′ 2 = dim Z r = dim X n − l. Take a basis β 1 , ..., β n−l of the space m ′ /m ′ 2 and let β i ∈ m/m 2 correspond to β i under the correspondence (2.1). Note that vectors β 1 , ..., β n−l , α 1 , ..., α l generate the space m/m 2 . This means that dim C m/m 2 ≤ n − l + l = n = dim X n . Since we also have the general inequality dim C m/m 2 ≥ dim X n , we obtain the equality dim C m/m 2 = dim X n , which means that X n is nonsingular at z.