Thom Isotopy Theorem for Nonproper Maps and Computation of Sets of Stratified Generalized Critical Values
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Abstract
Let \(X\subset {\mathbb {C}}^n\) be an affine variety and \(f:X\rightarrow {\mathbb {C}}^m\) be the restriction to X of a polynomial map \({\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\). We construct an affine Whitney stratification of X. The set K(f) of stratified generalized critical values of f can also be computed. We show that K(f) is a nowhere dense subset of \({\mathbb {C}}^m\) which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for nonproper polynomial maps on singular varieties.
Keywords
Isotopy lemma Affine varieties Nonproper polynomial mapping Local trivial fibrationMathematics Subject Classification
Primary 32B20 Secondary 14PXX1 Introduction
Ehresmann’s fibration theorem [3] states that a proper smooth surjective submersion \(f:X\rightarrow N\) between smooth manifolds is a locally trivial fibration. With some extra assumptions, this result has been considered in different contexts.
Firstly, if we remove the assumption of properness or submersiveness, in general, Ehresmann’s fibration theorem does not hold, since f might have “local singularities” or “singularities at infinity.” The set of points in N where f fails to be trivial, denoted by B(f), is called the bifurcation set of f, which is the union of the set \(K_0(f)\) of critical values and the set \(B_\infty (f)\) of bifurcation values at infinity of f. To date, characterizing \(B_\infty (f)\) remains an open problem. In general, a larger (but easier to describe) set is used, viz. the set of asymptotic critical values of f (see Definition 3.1), denoted by \(K_\infty (f)\), to control \(B_\infty (f)\). The set \(K_\infty (f)\) is always a nowhere dense subset of \({\mathbb {C}}^m\) and is a good approximation of the set \(B_\infty (f)\). For a dominant map \(f:X\rightarrow {\mathbb {C}}^m\) on a smooth complex affine variety X, the computation of \(K_\infty (f)\), and hence of the set of generalized critical values, \(K(f):=K_0(f)\cup K_\infty (f)\), is given in [8, 9, 10].
Now, if X is singular, one must partition it into disjoint smooth manifolds and then apply Ehresmann’s fibration theorem on each part. However, if we do not require any extra assumptions, then the trivialization on the parts may not match. This obstacle can be overcome by introducing the Whitney conditions [21, 22]. Indeed, if f is proper and X admits a Whitney stratification, then f is locally trivial if it is a submersion on strata [13, 18, 20]. Moreover, if f is nonproper and nonsubmersive, we can also define the bifurcation set of f such that f is locally trivial outside B(f). However, to date, to the best of the authors’ knowledge, no connection between B(f) and the set of stratified generalized critical values off, defined by \(K(f, \mathcal {S}):=\bigcup _{X_\alpha \in \mathcal {S}}K(f, {X_\alpha })\), for a Whitney stratification \(\mathcal {S}\) of X, has been established. Here \(K(f, X_\alpha )=K_0(f, X_\alpha )\cup K_\infty (f, {X_\alpha })\), where \(K_0(f, {X_\alpha })\) is the closure of the set of critical values of \(f_{{X_\alpha }}\) and \(K_\infty (f, {X_\alpha })=\{ y\in {\mathbb {C}}^m: \mathrm{there \ is \ a \ sequence} \ x^k\rightarrow \infty , \, x^k\in X_\alpha \text { such that } \Vert x^k\Vert \,\nu (\mathrm {d}_{x^k}(f_{X_\alpha }))\rightarrow 0\ \mathrm{and} \ f(x^k)\rightarrow y\}\) (\(\nu \) denotes the Rabier function, see Sect. 5).

Construct an affine Whitney stratification \(\mathcal {S}\) of X.

Establish some version of the Thom isotopy lemma for f which yields the inclusion \(B(f)\subset K(f, \mathcal {S})\).

Calculate the set \(K(f,\mathcal {S})\) of stratified generalized critical values of f.
2 Affine Whitney Stratifications
2.1 Preliminaries
For any two different points \(x,y\in {\mathbb {C}}^n\), define the secant \(\overline{xy}\) to be the line passing through the origin which is parallel to the line through x and y.
A stratification\(\mathcal {S}\) of X is a decomposition of X into a locally finite disjoint union \(X = \bigsqcup _{\alpha \in I} X_\alpha \) of nonempty, nonsingular, connected, locally closed subvarieties, called strata, such that the boundary \(\partial X_\alpha :={\overline{X}}_\alpha {\setminus } X_\alpha \) of any stratum \(X_\alpha \) is a union of strata. If, in addition, for each \(\alpha \), the closure \({\overline{X}}_\alpha \) and the boundary \(\partial X_\alpha \) are affine varieties, then we call \(\mathcal {S}\) an affine stratification. It is obvious that any affine stratification is finite.
 (a)
The pair \((X_\alpha ,X_\beta )\) is said to be Whitney (a) regular at\(x\in X_\beta \) if it satisfies the following Whitney condition (a) at x: if \(x^k\in X_\alpha \) is any sequence such that \(x^k\rightarrow x\) and \(T_{x^k}X_\alpha \rightarrow T\), then \(T\supset T_{x}X_\beta \).
 (w)
The pair \((X_\alpha ,X_\beta )\) is said to be (w) regular at \(x\in X_\beta \) (or strictly Whitney (a) regular atxwith exponent 1) if it satisfies the following condition (w) at x: there exist a neighborhood U of x in \({\mathbb {C}}^n\) and a constant \(c>0\) such that, for any \(y\in X_\alpha \cap U\) and \(x'\in X_\beta \cap U\), we have \(\delta (T_{x'}X_\beta ,T_yX_\alpha )\leqslant c\Vert yx'\Vert \).
 (b)
The pair \((X_\alpha ,X_\beta )\) is said to be Whitney regular at\(x\in X_\beta \) if it satisfies the following Whitney condition (b) at x: for any sequences \(x^k\in X_\alpha \) and \(y^k\in X_\beta \), \(y^k\ne x^k\), such that \(x^k\rightarrow x\), \(y^k\rightarrow x\), \(T_{x^k}X_\alpha \rightarrow T\), and \(\overline{x^ky^k}\) converges to a line \(\ell \) in the projective space \({\mathbb {P}}^{n1}\), we have \(\ell \subset T\).
For the purpose of this paper, we also need the following notion of Whitney (resp. Whitney (a)) regularity along a stratum: Let \(X_\beta \) be a stratum of \(\mathcal {S}\) and let \(x\in X_\beta \). We say that \(X_\beta \) is Whitney regular (resp. Whitney (a) regular) at x if, for any stratum \(X_\alpha \) such that \(X_\beta \subset {\overline{X}}_\alpha \), the pair \((X_\alpha ,X_\beta )\) is Whitney (resp. Whitney (a)) regular at x. The stratum \(X_\beta \) is Whitney regular (resp. Whitney (a) regular) if it is Whitney (resp. Whitney (a)) regular at every point of \(X_\beta \). It is clear that \(\mathcal {S}\) is a Whitney (resp. a Whitney (a)) stratification if and only if each stratum of \(\mathcal {S}\) is Whitney (resp. Whitney (a)) regular.
2.2 Construction of Affine Stratifications
Let us, first of all, fix an affine stratification of X whose construction is based on the following proposition:
Proposition 2.1
Let \(X\subset {\mathbb {C}}^n\) be an affine subvariety of pure codimension r. Assume that \(I(X)=(g_1,\ldots ,g_\omega )\), where \(\deg g_i\le D\). Let W be an affine subvariety of positive codimension in X with \(I(W)=(g_1,\ldots ,g_\omega , u_1,\ldots ,u_\tau )\), where \(u_i\not \in I(X)\) and \(\deg u_i\leqslant D'\). Then there exists a polynomial \(p_{X,W}\) on \({\mathbb {C}}^n\) of degree less than or equal to \(r(D1)+D'\) such that \(W\subseteq V(p_{X,W}):=\{x\in {\mathbb {C}}^n:\ p_{X,W}(x)=0\}\) and \(X{\setminus } V(p_{X,W})\) is a smooth, dense subset of X. Moreover, the polynomial \(p_{X,W}\) can be constructed effectively.
Proof
Remark 2.2
Theoretically, a random rational number is a generic rational number, but practically by random numbers we mean rational numbers produced by special random algorithms.
The polynomial \(p_{X,W}\) can be found by using a probabilistic algorithm. First recall the following:
Definition 2.3
Let I be an ideal in \({\mathbb {C}}[x_1,\ldots ,x_n]\). We define the homogenization of I to be the ideal \(I^\mathrm {h}\) generated by \(\{f^\mathrm {h}: f\in I \}\subset {\mathbb {C}}[x_0,\ldots ,x_n]\), where \(f^\mathrm {h}\) is the homogenization of f.
Theorem 2.4
([2, Thm. 4, §4, Chap. 8, p. 388]) Let I be an ideal in \(k[x_1,\ldots ,x_n]\) and let \(G=\{g_1,\ldots ,g_s\}\) be a Gröbner basis for I with respect to a graded lexicographic order in \(k[x_1,\ldots ,x_n]\) (i.e., the lexicographic order that first compares the total degree: \(x^\alpha > x^\beta \) whenever \(\alpha  > \beta \)). Then \(G^\mathrm {h}=\{g^\mathrm {h}_1,\ldots ,g^\mathrm {h}_s\}\) is a basis for \(I^\mathrm {h}\subset k[x_0,x_1,\ldots ,x_n]\).
This theorem allows us to compute the set of points at infinity of an affine variety given by the ideal I; to this aim, it is enough to compute the Gröbner basis \(\{g_1,\ldots ,g_s\}\) of the ideal I and then to consider the ideal \(I_\infty =(x_0, g^\mathrm {h}_1,\ldots ,g^\mathrm {h}_s)\). In particular, we can check effectively whether \(L^r\cap \overline{X}\cap \{x_0=0\}=\emptyset \), which implies that Open image in new window for \(i=1,\ldots ,m\) (see the proof of Proposition 2.1). This is crucial for our computations.

INPUT: The ideal \(I=I(X)=(g_1,\ldots ,g_\omega )\) and the ideal \(J=I(W)=(g_1,\ldots ,g_\omega ,u_1,\ldots ,u_\tau )\)

1) repeat

choose random rational numbers \(\alpha _{i1},\ldots ,\alpha _{i\omega }\), \(i=1,\ldots ,r\);

put \(G_i:=\sum ^\omega _{k=1} \alpha _{ik} g_k\), \(i=1,\ldots ,r\);

put \(I=( G_1,\ldots , G_r)\);

until\(\dim V(I)=nr\).

2) repeat

choose random rational numbers \(\beta _{i1},\ldots ,\beta _{in}\), \(i=1,\ldots ,nr\);

put \(l_i:=\sum ^n_{k=1} \beta _{ik} x_k\), \(i=1,\ldots ,nr\);

put \(I=( G_1,\ldots , G_r,l_1,\ldots ,l_{nr})\);

compute the ideal \(I_\infty =(H_1,\ldots ,H_m)\subset k[x_0,\ldots ,x_n]\);

if dim \(V(I_\infty )=0\)then

begin

compute \(V(G_1,\ldots ,G_r, l_1,\ldots ,l_r):=\{a_1,\ldots ,a_p\}\)

end

until\(\dim V(I_\infty )=0\) and \(\mathrm{Jac}(G_1,\ldots ,G_r,l_1,\ldots ,l_{nr})(a_i)\ne 0\) for \(i=1,\ldots p\).

3) repeat

choose random rational numbers \(\gamma _{1},\ldots ,\gamma _{\tau }\);

put \(H:=\sum ^\tau _{k=1} \gamma _{i} u_k\) ;

put \(J=( G_1,\ldots , G_r, H)\);

until\(\dim V(J)<nr\).

OUTPUT: \(p_{X,W}=\mathrm{Jac}(G_1,\ldots ,G_r,l_1,\ldots ,l_{nr})\cdot H\)
Remark 2.5
Let us assume that I(X) and I(W) are generated by polynomials from the ring \({\mathbb {F}}[x_1,\ldots ,x_n]\), where \({\mathbb {F}}\) is a subfield of \({\mathbb {C}}\). Then we can choose a polynomial \(p_{X,W}\) such that \(p_{X,W}\in {\mathbb {F}}[x_1,\ldots ,x_n]\).
From the proof of Proposition 2.1, with no loss of generality, we can assume that \({\mathrm{rank}}\,\mathrm{Jac}(g_1,\ldots ,g_r) = r\) on some nonempty regular open subset \(X^0\) of X and that \(X=\overline{ X^0}\). It is clear that \(V(p_{X,W})\) contains \({\mathrm{sing}}(X)\cup W\) and the singular points of the projection \((l_1,\ldots ,l_{nr}):X\rightarrow {\mathbb {C}}^{nr}\). Now, to construct an affine stratification of X, it is enough to construct an affine filtration \(X=X_0\supset X_1\supset \cdots \supset X_{nr}\supset X_{nr+1}=\emptyset \) by induction with \(X_{i+1}:=X_i\cap V(p_{X_i,\emptyset }),\ i=0,\ldots ,nr\). The degree of each \(X_i\) can be calculated and depends only on D.
2.3 Construction of Affine Whitney Stratifications
In this section, we construct an affine Whitney stratification of a given affine variety X, with \(I(X)=(g_1,\ldots ,g_r)\) and \(\deg g_i\le D\), by refining the affine stratification given in Sect. 2.2 so that the resulting stratification is still affine and moreover satisfies the Whitney condition.
Lemma 2.6

\(x^k\rightarrow x\),

\(y^k\rightarrow x\),

\(w^k:=\gamma ^k(x^ky^k)\rightarrow w\),

\(v^k:=\sum _{i=1}^r\lambda ^k_i\mathrm {d}_{x^k} g_i\rightarrow v\).
Proof

either \({\bar{x}}^k=y^k\) for every k or \({\bar{x}}^k\ne y^k\) for every k,

for each i, either \(\lambda ^k_i\ne 0\) for every k or \(\lambda ^k_i=0\) for every k.
The following algebraic criterion permits us to check Whitney regularity on \(Y^0=Y{\setminus } V(p_{Y,W})\), where the notation \(V(p_{Y,W})\) is from Proposition 2.1 and the affine set W will be determined later.
Lemma 2.7
Let \(x\in Y^0\). Then the pair \((X^0,Y^0)\) satisfies the Whitney condition (b) at x if and only if, for any \((x,x,w,v)\in C(X,Y)\), we have \( v\cdot w=0\).
Proof

\(x^k\rightarrow x\), \(y^k\rightarrow x\),

\(w^k:=\gamma ^k(x^ky^k)\rightarrow w\),

\(v^k:=\sum _{i=1}^r\lambda ^k_i\mathrm {d}_{x^k} g_i\rightarrow v\).

\(x^k\ne y^k\), \( x^k\rightarrow x\), \( y^k\rightarrow y\);

\(T_{x^k}X^0\rightarrow T\);

the sequence of secants \(\overline{x^ky^k}\) tends to a line \(\ell \not \subset T\).
Lemma 2.8
Proof
Note that \(x\in Y^0\) if and only if \(p_{Y,\emptyset }(x)\ne 0\), i.e., there exists \(\lambda \in {\mathbb {C}}\) such that \(\lambda p_{Y,\emptyset }(x)=1\). In view of Lemma 2.7, the pair \((X^0,Y^0)\) is not Whitney regular at x if and only if there exist w, v with \(v\cdot w\ne 0\) such that \((x,x,w,v)\in C(X,Y)\). The lemma follows easily. \(\square \)

\(X_0:=X\),

\(X_1:=X_0\cap V(p_{X_0,\emptyset })\),

\(X_2:=X_1\cap V(p_{X_1,W(X_0,X_1)})\),

\(X_3:=X_2\cap V(p_{X_2,W(X_0,X_2)\cup W(X_1,X_2)})\), \(\ldots \),

\(X_i:=X_{i1}\cap V\bigl (p_{X_{i1},\bigcup _{j=0}^{i2} W(X_j,X_{i1})}\bigr )\), \(\ldots \)
3 Thom Isotopy Lemma for Nonproper Maps
We start this section with:
Definition 3.1
By [10, Thm. 3.3], we have that, for every \(\alpha \), the set \(K_\infty (f_{X_\alpha })\) is closed and has measure 0 in \({\mathbb {C}}^m\). In particular the set K(f) defined below is also closed and has measure 0.
Definition 3.2
Remark 3.3

\(\Vert f(x^{kj_k})y^k\Vert < 1/k\),

\(\Vert x^{kj_k}\Vert >k\),

\(\Vert x^{kj_k}\Vert \,\nu (\mathrm {d}_{x^{kj_k}} f)< 1/k\).
Assuming that \(\mathcal {S}\) is an affine Whitney stratification of X, we prove that \(K(f,\mathcal {S})\) contains the set of bifurcation values of f.
Theorem 3.4
(First isotopy lemma for nonproper maps) Let \(X\subset {\mathbb {C}}^n\) be an affine variety with an affine Whitney stratification \(\mathcal {S}\), and let \(f=(f_1,\ldots ,f_m):X\rightarrow {\mathbb {C}}^m\) be a polynomial dominant map. Let \(K(f,\mathcal {S})\) be the set of stratified generalized critical values of f given by (2). Then f is locally trivial outside \(K(f,\mathcal {S})\).

The restriction \(\psi _{X_\beta }\) to any stratum \(X_\beta \) is a smooth function.

For any stratum \(X_\beta \) and for any \(x\in X_\beta \), there exist a neighborhood U of x in \({\mathbb {R}}^{2n}\) and a constant \(c>0\) such that, for any \(y\in X\cap U\) and \(x'\in X_\beta \cap U\), we have \(\psi (y)\psi (x')\leqslant c\Vert yx'\Vert \).
Proof of Theorem 3.4
Lemma 3.5
For \(i=1,\ldots ,2m\), let \(v_i\) be vector fields on X which are rugose in U. Assume that \(\mathrm {d}f(v_i)=\partial _i\) and there is a positive constant \(c>0\) such that \(\Vert v_i(x)\Vert \leqslant \frac{\Vert x\Vert +1}{c}\) for any \(x\in U\). Then f is a topologically trivial fibration over B.
Proof
 (i)
The flow of \(v_i\) preserves the stratification. This is a consequence of the rugosity. For more detail, see [20, Prop. 4.8].
 (ii)
For each i and any \(x\in U\), there is a unique integral curve of \(v_i\) passing through x. This follows from the uniqueness of integral curves of smooth vector fields and the fact that \(v_i\) preserves the stratification.
Claim 3.6
For each \(x\in f^{1}(Y^i_0)\), let \(\gamma \) be the integral curve of \(v_i\) such that \(\gamma (0)=x\). Then \(\gamma \) reaches any level \(f^{1}(Y^i_t)\) at time t for \(t\in (1,1)\).
Proof
Now let us prove the following:
Lemma 3.7
Proof
Assume for contradiction that there exist an index \(\beta \in I'\) and a sequence \(x^k\in {\overline{U}} \cap X_\beta \) such that \((\Vert x^k\Vert +1)\,\nu (\mathrm {d}_{x^k}(f_{X_\beta }))\rightarrow 0\). Taking a subsequence if necessary, we can suppose that \(x^k\rightarrow x\), \( T_{x^k}X_\beta \rightarrow T\) and \(f(x^k)\rightarrow y\in {\overline{B}}\) with \(x\in {\mathbb {C}}^n\) or \(x=\infty \). If \(x=\infty \), then by definition, \(y\in K_\infty (f_{X_\beta })\subset K(f,\mathcal {S})\). This is a contradiction since \({\overline{B}}\cap K(f,\mathcal {S})=\emptyset \). Thus \(x\in {\mathbb {C}}^n\), and we get \(\nu (\mathrm {d}_{x^k}(f_{X_\beta }))\rightarrow 0\). In the case \(x\in X_\beta \), in view of [15, Lem. 2.2], we have \(\nu (\mathrm {d}_{x}(f_{X_\beta }))= 0\), i.e., \(y\in K_0(f,X_\beta )\), which is also a contradiction. Therefore \(x\in {\overline{X}}_\beta {\setminus } X_\beta \). Denote by \(X_\alpha \) the stratum containing x. Let \(F=(f_1,\ldots ,f_m):{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m\) be the polynomial extending f on \({\mathbb {C}}^n\). Obviously \(\nu (\mathrm {d}_{x^k}F_{T_{x^k}X_\beta })=\nu (\mathrm {d}_{x^k}(F_{X_\beta }))=\nu (\mathrm {d}_{x^k}(f_{X_\beta }))\rightarrow 0\). Moreover, since \(f_{X_\alpha }\) is a submersion at x, so F is a submersion at x. Hence \(\nu (\mathrm {d}_{x}(F_{X_\alpha }))\ne 0\). Since the stratification is Whitney, it implies that \(T\supset T_xX_\alpha \). Consequently \(\nu (\mathrm {d}_{x}F_{T})\geqslant \nu (\mathrm {d}_{x}(F_{X_\alpha }))\ne 0\). To get a contradiction, we will need the following claim:
Claim 3.8
Let \(A_k:{\mathbb {R}}^q\rightarrow {\mathbb {R}}^p\) be a sequence of linear maps such that \(A_k\rightarrow A\) as \(k\rightarrow +\infty \) (i.e., the terms of the matrix of \(A_k\) tend to the corresponding terms of the matrix of A). Let \(H_k\subset {\mathbb {R}}^q\) be a sequence of linear subspaces of same dimension such that \(H_k\rightarrow H\) (i.e., \(\delta (H_k,H)\rightarrow 0\), where \(\delta (H_k,H):=\sup _{y\in H_k,\Vert y\Vert =1}{\mathrm{dist}}(y,H)\) is the distance between \(H_k\) and H; \({\mathrm{dist}}(\cdot ,\,\cdot \,)\) is the Euclidean distance). Then \(\nu (A_k_{H_k})\rightarrow \nu (A_H)\).
Proof
Lemma 3.9
Proof
Note that, for fixed i, the vector field on U which coincides with \(v_i^\beta \) on each \(U_\beta \) is not necessarily a rugose vector field. In what follows, we will try to deform these vector fields to produce a rugose vector field which satisfies the assumption of Lemma 3.5. The process is carried out by induction on dimension.
For \(2m\leqslant d\leqslant 2\dim _{\mathbb {C}}X\), let \(I'_d:=\{\beta \in I':2m\leqslant \dim X_\beta \leqslant d\}\) and \(U_d:=\bigcup _{\beta \in I'_{d}}X_\beta \cap U\). By induction on d, we construct a rugose vector field on \(U_{2\dim _{\mathbb {C}}X}\) with the property of Lemma 3.5. For \(d=2m\), let \(v_i^{2m}\) be the restriction to \(U_{2m}\) of the smooth vector field on \(\bigcup _{\beta \in I'_{2m}}X_\beta \) which coincides with each \(v_i^\beta \) on \(X_\beta \) for \(\beta \in I'_{2m}\). Then \(v_i^{2m}\) is clearly rugose, \(\mathrm {d}f(v_i^{2m})=\partial _i\) and by Lemma 3.9, \(\Vert v_i^{2m}(x)\Vert \leqslant \frac{\Vert x\Vert +1}{c}\) for any \(x\in U_{2m}\).
For each i, assume that we have constructed a rugose vector field, denoted by \(v_i^d\), on \(U_d\) such that \(\mathrm {d}_xf(v_i^{d}(x))=\partial _i\) and \(\Vert v_i^d(x)\Vert \leqslant \frac{\Vert x\Vert +1}{c_d}\) for every \(x\in U_d\), where \(c_d\) is a positive constant. We need to extend each \(v_i^d\) to a rugose vector field \(v_i^{d+2}\) on \(U_{d+2}\) such that \(\Vert v_i^{d+2}(x)\Vert \leqslant \frac{\Vert x\Vert +1}{c_{d+2}}\) for every \(x\in U_{d+2}\), where \(c_{d+2}\) is also a positive constant (recall that the strata of \(\mathcal {S}\) have even dimension). Note that, to construct \(v_i^{d+2}\), it is enough to construct \(v_i^{d+2}\) separately on each stratum \(X_\alpha \) with \(\alpha \in I'_{d+2}{\setminus } I'_d\). Without loss of generality, suppose that \(I'_{d+2}{\setminus } I'_d=\{\alpha \}\). By [20, Lem. 4.4], for each \(i=1,\ldots ,2m\), there is a rugose vector field on \(U_{d+2}\), denoted by \({\widetilde{w}}_i^{d+2}\), which extends \(v_i^d\), so the restriction \({\widetilde{w}}_i^{d+2}_{U_{d+2}\cap X_\alpha }\) is a smooth vector field. We need to adjust \({\widetilde{w}}_i^{d+2}\) to get a new rugose vector field \(w_i^{d+2}\) on \(U_{d+2}\) such that, for any \(y\in X_\alpha \cap U_{d+2}\), we have \(\mathrm {d}_yf(w_i^{d+2}(y))=\partial _i\).
Lemma 3.10
Proof

\(\Vert P(y)Q(x',y)\Vert <C\Vert yx'\Vert \),

\(\Vert S(x',y)\Vert <C\Vert yx'\Vert \)
 Since \(1\varphi (y)\) is a smooth function, it is locally Lipschitz; with no loss of generality, assume that \(1\varphi (y)\) is Lipschitz on \(W_x\) with constant \(c_1\). Then$$\begin{aligned} 1\varphi (y)=(1\varphi (y))(1\varphi (x'))\leqslant c_1\Vert yx'\Vert . \end{aligned}$$

By Lemma 3.9 and by the continuity of \(w_i^{d+2}\), there is a positive constant \(c_2\) depending only on x such that \(\Vert v_i^\alpha (y)\Vert +\Vert w_i^{d+2}(y)\Vert \leqslant c_2\) (we can take \( c_2:=\sup _{z\in {\overline{W}}_x\cap X_\alpha }\frac{\Vert z\Vert +1}{c}+\sup _{z\in {\overline{W}}_x\cap X_\alpha }\Vert w_i^{d+2}(z)\Vert \); note that \({\overline{W}}_x\cap X_\alpha \subset U_{d+2}\)).

Since \(w_i^{d+2}\) is rugose, it follows that there is a positive constant \(c_3\) depending only on x such that \(\Vert w_i^{d+2}(y)w_i^{d+2}(x')\Vert \leqslant c_3\Vert yx'\Vert \).
The following corollary follows immediately from Theorem 3.4:
Corollary 3.11
Let \(X\subset {\mathbb {C}}^n\) be an affine variety with an affine Whitney stratification \(\mathcal {S}\), and let \(f:X\rightarrow {\mathbb {C}}^m\) be a polynomial dominant map. Assume that, for any stratum \(X_\beta \in \mathcal {S}\), the restriction \(f_{X_\beta }\) is a submersion and \(K_\infty (f, {X_\beta })=\emptyset \). Then f is a locally trivial fibration.
4 Computation of the Sets of Stratified Generalized Critical Values

As the construction of Whitney stratifications is by induction on dimension, we only need to proceed until the dimension shrinks below m, since the restriction of f to any stratum of dimension \(<m\) is always singular.
 For any algebraic set \(Z\subseteq X\), letThen, at any step of the induction process, the construction in Sect. 2.3 can be omitted if \(r_Y<m\).$$\begin{aligned}&\displaystyle r_Z:=\max _{x\in Z{\setminus } V(p_{Z,\emptyset })}{\mathrm{rank}}\,\mathrm{Jac}_x(f_Z)\\&\quad \text { and }\ H(Z):=\overline{\{x\in Z{\setminus } V(p_{Z,\emptyset }):{\mathrm{rank}}\,\mathrm{Jac}_x(f_Z)<r_Z\}}{}^{\,\mathcal {Z}}. \end{aligned}$$

\(X_0:=X\),

\(X_1:=X_0\cap V(p_{X_0,\emptyset }),\ S_1=K_0(f, {X_0{\setminus } X_1}), \ldots \),

\(X_i:=X_{i1}\cap V(p_{X_{i1},\bigcup _{j=0}^{i2} {\widetilde{W}}(X_j,X_{i1})}),\ S_i=K_0(f, {X_{i1}{\setminus } X_i}), \ldots \)
Corollary 4.1
Let \(X\subset {\mathbb {C}}^n\) be an affine variety of pure dimension and let \(f=(f_1,\ldots , f_m) :X\rightarrow {\mathbb {C}}^m\) be a polynomial mapping. Let \({\mathbb {F}}\subset {\mathbb {C}}\) be a subfield generated by coefficients of generators of I(X) and all coefficients of polynomials \(f_i, \ i=1,\ldots ,m\). Then there is a nowhere dense affine variety \(K(f,\mathcal {S})\subset {\mathbb {C}}^m\), which is described by polynomials from \({\mathbb {F}}[x_1,\ldots ,x_m]\) such that all bifurcation values B(f) of f are contained in \(K(f,\mathcal {S})\). In particular, for \(m=1\), if X and f are described by polynomials from \({\mathbb {Q}}[x_1,\ldots ,x_n]\), then all bifurcation values of f are algebraic numbers.
5 Computation of \(K_0(f, {Z_i})\cup K_\infty (f, {Z_i})\)
Let \(k={\mathbb {R}}\) or \(k={\mathbb {C}}\). Let \(X\cong k^n\), \(Y\cong k^m\) be finitedimensional vector spaces (over k). We consider those spaces equipped with the canonical scalar (hermitian) products. Let us denote by \({{\mathcal {L}}}(X, Y)\) the set of linear mappings from X to Y and by \(\Sigma =\Sigma (X,Y)\subset {{\mathcal {L}}}(X, Y)\) the set of nonsurjective mappings. In this section, we give several different expressions for a distance of an \(A\in {{\mathcal {L}}}(X, Y)\) to the space \(\Sigma \) of singular operators. Let us first recall the following [15]:
Definition 5.1
Proposition 5.2
Definition 5.3
From Proposition 5.2 we immediately get the following corollary:
Corollary 5.4
We have \(\nu (A, H) \sim \kappa (A, H)\).
In fact we also have the following explicit expression for \(\kappa (A, H)\) (see [9, 10]):
Proposition 5.5
Finally, we introduce a function \(g'\) which will be useful in the explicit description of the set of generalized critical values.
Definition 5.6
In particular, we have the following (see [9, 10]):
Proposition 5.7
We have \(g'(A, H)\sim \nu (A, H)\).
Now we can prove the following theorem:
Theorem 5.8
Let \(Z_i\) be a stratum of X as in Sect. 4. Then the set \(K(f, {Z_i})=K_0(f, {Z_i})\cup K_\infty (f, {Z_i})\) is a nowhere dense algebraic subset of \({\mathbb {C}}^m\).
Proof
It is a standard fact that \(K_0(f, {Z_i})\) is algebraic and nowhere dense (for details see the end of Sect. 5.1). Hence, it is enough to focus on \(K_\infty (f, {Z_i})\).
By construction, the set \(X:= Z_i\subset {\mathbb {C}}^{n}\) is a subset of complete intersection, \(X \subset \{ b_1=0,\ldots , b_s=0\}\), and \({\mathrm{rank}}\{\nabla b_k: k=1,\ldots ,s\}=s\) (X has codimension s). Let us recall notation of Definition 5.6. For \(x \in {\mathbb {C}}\), let \(A=\mathrm {d}_x f\) and \(B_l = \mathrm {d}_x b_l \), \(l=1,\ldots , s\). Let \(A\in {{\mathcal {L}}}(k^n, k^m)\), where \(n\ge m+s\), and let \(T_xX =H\subset k^n\) be a linear subspace given by a system of independent linear equations \(B_l=\sum b_{lk} x_{k}, \, l=1, \ldots , s\). By abuse of notation, we denote by A the matrix (in the canonical bases in \(k^n\) and \(k^m\)) of the mapping A. Let C be an \((m+s)\times n \) matrix given by rows \(A_1,\ldots ,A_m; B_1,\ldots ,B_s\) (we identify \(A_i=\sum a_{ik} x_k\) with the vector \((a_{j1},\ldots , a_{jn})\), similarly for \(B_l\)).
For an index \(I=(i_1,\ldots ,i_{m+r})\subset \{1,\ldots , n\}\) let \(M_{I}(x)\) denote the \((m+s)\times (m+s)\) minor of C given by columns indexed by I. For integers \(j\in I\), \(1\le k\le m\) we denote by \(M_{I(k,j)} (x)\) the \((m+s1)\times (m+s1)\) minor obtained by deleting the \(j^\text {th}\) column and the \(k^\text {th}\) row. Note that we delete only \(A_k\), \(1\le k\le m\) rows.
We can assume that, for some choice of l, we have \(W_{I_l(k_l,j_l)}\not \equiv 0\), and consequently \(\dim \mathrm{cl}(\Phi _{(k,j)}(X))=\dim X = ns\). Here \(\mathrm{cl}(Y)\) stands for the closure of Y in the strong (or which is the same, in the Zariski) topology. Let \(\Gamma (k,j)= \mathrm{cl}( \Phi _{(k,j)}(X))\) (by \(\Phi _{(k,j)}(X)\) we mean the set \(\Phi _{(k,j)}(X{\setminus } P)\), where P is a set of poles of \(\Phi _{(k,j)}\)).
Lemma 5.9
Proof
We identify X with \({\widetilde{Z}}_i\subset {\mathbb {C}}^{n+1}\), hence we can assume that X is closed in \({\mathbb {C}}^{n+1}\). Let \(y\in K_\infty (f,X)\). Hence, there is a sequence \(x^l\rightarrow \infty \) such that \(x^l\in X\) and \(f(x^l)\rightarrow y\) and \(\Vert x^l\Vert \, g'(x_l) \rightarrow 0\). Moreover, if \(x=(x_1,\ldots ,x_n)\), then there is at least one r, \(1\le r\le n\), such that \(x^l_r\rightarrow \infty \). If \(\{x^l: l=1,2,\ldots \} \subset C(f,X)\) (C(f, X) denotes the set of critical points of \(f_{X}\)), then it is easy to see that \(y\in {\mathbb {C}}^m \cap \Gamma ((k,j),r)\) for every (k, j) (we can choose a close sequence \(x'^l\) such that \( f(x'^l)\rightarrow y\) and \(\Vert x'^l \Vert \, g'(x'^l) \rightarrow 0\) and functions \( W_{I(k,j)}\) are defined). Consequently, we can assume that \(\{x^l: l=1,2,\ldots \}\cap C(f, X)=\emptyset \). Thus, there is a sequence \(x^l\rightarrow \infty \) such that, for every \(I_i\), there are integers \((k_i,j_i)\) such that \(\Vert x^l\Vert \, M_{I_i}/M_{I_i(k_i,j_i)}(x_l)\rightarrow 0\) and \(f(x^l)\rightarrow y\). This also gives \(y\in \Gamma ((k,j),r) \cap {\mathbb {C}}^m\) with \(((k,j),r) = ((k_1,j_1),\ldots ,(k_w,j_w),r)\).
Conversely, if \(y\in \Gamma ((k,j),r)\cap {\mathbb {C}}^m\), then we can choose a sequence \(x^l\rightarrow \infty \), \(x^l\in X_r\) such that \(f(x^l)\rightarrow y\) and \(\Vert x^l\Vert \,M_{I_i}/M_{I_i(k_i,j_i)}(x^l)\rightarrow 0\). It is easy to observe that this implies \(\Vert x^l\Vert \,g'(x^l)\rightarrow 0\) and \(f(x^l)\rightarrow y\), i.e. \(y\in K_\infty (f, X)\). \(\square \)
Now, in light of [10, Thm. 3.3], we have that \(K_\infty (f,X)\ne {\mathbb {C}}^m\) hence \({\mathbb {C}}^m \cap \bigcup _{((k,j),r)} \Gamma ((k,j),r)\ne {\mathbb {C}}^m\). By Lemma 5.9, \(K_\infty (f,X)\) is an algebraic set. The theorem follows. \(\square \)
5.1 A Sketch of an Algorithm
Let \(X:=Z_i\subset {\mathbb {C}}^{n}\) be a smooth affine variety of dimension \(ns\). Let \(f=(f_1,\ldots , f_m):X \rightarrow {\mathbb {C}}^m\) be a polynomial dominant mapping. Then the set \(K_\infty (f,X)\) can be computed as follows:
Notes
Acknowledgements
We would like to thank Nguyen Xuan Viet Nhan and Nguyen Hong Duc for helpful discussion during the preparation of this paper. We would also like to thank the referees for their careful reading, valuable comments, and suggestions. S. T. Ðinh is partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.042017.12. Z. Jelonek is partially supported by Narodowe Centrum Nauki grant no. 2015/17/B/ST1/02637.
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