By H. T. F. Braun
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Rd−1 using the holomorphic selection of y1 , . . , yd . The representation depends on the partition s; I write out just two cases. Suppose for example that s = (1, . . , 1) is the partition into d points, and that X is a member of the cover of Vs found above. So y1 (x), . . , yd (x) ∈ OUR are holomorphic selections of the zeros of f on X; for each x ∈ U they are all distinct. 2) at each (x, yi ) to establish the corresponding row of the relation R0 (x) g(x, y1 (x)) 1 y1 · · · y1d−1 ..
Am ) ∈ U Partial derivatives and analytic continuation 46 be a point of the domain of f : U → C, and let e ∈ (Cm \ {0}) be a non-zero vector. Then we shall say that f is regular at a with respect to e if there is a linear change of basis φ : Cm → Cm such that φ : (0, . . , 0, 1) → e and the function f (x1 , . . , xm ) = f ◦ φ−1 (x1 − a1 , . . , xm − am ) is regular in xm . We will exploit two results from the theory of o-minimal expansions of the real field. 3 (Good Directions Lemma, van den Dries [19]) Let A ⊆ Rn+1 be definable in an o-minimal expansion R of an ordered field, with dim(A) ≤ n; fix a metric · .
Then (a , gj (a )) is a limit of points (ai , gj (ai )) in Ad but also of points (ai , bi + gj (ai )) which are zeros of f not in Cj . So f (a , y) has a zero of order strictly greater than d at y = gj (a ), or is not regular in y. So, as Cj ⊆ Ad , (a , gj (a )) ∈ Cj . We can therefore find an Ran -definable continuous function ε : π(Cj ) → R such that 0 < ε(a) < ε0 (a) for each a ∈ π(Cj ). (For example, for some n there is a definable homeomorphism σ : π(Cj ) → (−1, 1)n = {t ∈ Rn : t < 1}, the open unit box in Rn with l∞ norm.