Lemma 56 ([Uniform box data]). After the preceding finite subdivision, each nonzero comparable-scale field has, on a fixed larger state disk D and a smaller real interval I D , a positive globally subanalytic scale L(p,x) and constants K,m …

Normalize the denominator by its value at the disk center. In its factored representation its coefficients and its reciprocal are bounded on a slightly smaller disk, because all poles are uniformly separated. Let L_0 be the maximum modulus of the coefficients of the resulting numerator. Separate L_0=0 ; otherwise divide the numerator by L_0 . The normalized numerator and denominator belong to compact coefficient sets, the denominator is uniformly nonzero, and at least one numerator coefficient has modulus one. If the numerator degree is at most m , no member can have its first m+1 jets vanish at a point: multiplication by the nonzero denominator would give a polynomial of degree at most m with a zero of order m+1 . Compactness, including z∈I , gives a positive uniform lower bound on the sum of those jets. Rescaling L_0 by that fixed bound gives Equation. The supremum estimate follows from the coefficient bounds. All the normalizations use subanalytic operations and nonzero divisors.