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By Trjitzinsky W. J.

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We may also consider the tameness in the sense of Broughton [Br2], namely: ∃K > 0 such that, for any sequence |xi | → ∞, we have | grad f (xi )| ≥ K. Then ‘f is tame in the sense of Broughton’, obviously implies ‘f is tame at t, for any t ∈ C’, whereas the reciprocal is not true; some examples can be found in [NZ, Du2]. Remarks on polynomials with at most one atypical value. We have seen that, if a polynomial function has no critical points and all its fibres are t-regular at infinity, then there are no atypical values.

3 in the Appendix. 3) is a locally trivial C ∞ fibration. The isotopy type of the fibration does not depend on the choice of M and δ. 4) is a trivial C ∞ fibration. Proof Since the fibre f −1 (t0 ) might be singular, let us fix a complex semialgebraic Whitney stratification on it. By the Tarski–Seidenberg principle, all the levels of the distance function ρE , except for finitely many, are transversal to the strata of f −1 (t0 ). This means that there exists a sufficiently large value M0 > 0 such that the level ρE = M is transversal to f −1 (t0 ), for any M ≥ M0 .

Let us consider the map germ: (x0 , τ ) : (XK ∩ (Ui × K), y) → (K2 , 0), for some y ∈ X∞ K in the chart Ui ×K. Then consider the singular loci of the map (x0 , τ ) and of the restriction of τ to the nonsingular subspace Kn ∩ (Ui × K) and take the analytic closure of their difference: (x0 , τ )(i) y := closure[Sing (x0 , τ )|Kn ∩(Ui ×K) \ Sing τ|Kn ∩(Ui ×K) ] ⊂ X, viewed as a germ at y. 1 The germ at y ∈ X∞ K of the analytic set (x0 , τ )y is called local polar locus of τ with respect to x0 . It depends on the affine chart Ui × K since the function x0 does so.

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