By Gregory V. Chudnovsky

This quantity involves a set of papers dedicated essentially to transcendental quantity thought and diophantine approximations written by way of the writer. many of the fabrics integrated during this quantity are English translations of the author's Russian manuscripts, generally rewritten and taken completely brand new. those papers and different papers incorporated during this quantity have been to be had to experts in manuscript shape, yet this can be the 1st time that they've been accumulated and released. even though the sooner papers were preserved within the shape during which they have been ready before everything, the amount is geared up in one of these means as to mirror fresh development and to permit readers to stick with contemporary advancements within the box. As an introductory consultant to the quantity, the writer integrated an elevated and up-to-date textual content of his invited deal with on his paintings at the idea of transcendental numbers to the 1978 foreign Congress of Mathematicians in Helsinki. The appendix encompasses a paper at the extremality of definite multidimensional manifolds ready through A. I. Vinogradov and the writer in 1976. Chudnovsky bought a MacArthur origin Fellowship in 1981.

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Let A, Ax(z)9 A2(z)9 B(z) be as above. If a is algebraic a =£ 0, P(x9 y) G Z[x, y], P z 0, d(P) < d, H(P) < H, then (24) | P(Ax(a90)9 A2(a,0)) \> H~c*d2 for log log H > c21d5. Theorem 6 can be supplied with an additional normality statement (cf. [C14]). THEOREM 14. If p(z) has complex multiplication and u is an algebraic point of p(z)9 then \P{u9${u))\>H-c^ with c2S(d) > 0 depending on d and u9 p(z) only. Theorem 14 for example implies that any number P(u9 f(w)) is normal. As a corollary (or, equally, as a corollary of Theorem 3), values of the T-function T(l/3), T(l/4), etc.

Consider two polynomials R1 and R 2 with known heights and two corresponding divisors Px and P2 such that | P,(0) | and | P2(0) \ are small. Suppose P{ and P2 are coprime. What lower bound can we obtain for max{| P^O) \ , | P2(Q) |}? 5, then for In H >• n we obtain a very good estimate. Otherwise, because of the factor en attached to H(Pt), we obtain an unsatisfactory estimate. How can we improve it? 3, HP < HR. We now present several lemmas which are completely sufficient for our purposes, but which are not the best possible results.

Lntf)} where deg(Q(z))e its roots. 24) min \0-Zl\-e-nlH-'l<\P(Q)\ \e-^\-n2H(H min < i'•= 1 , . . , 2)"~lmax{\,\e\n}. 10 is given in [6]; it makes use of properties of the discriminant D — D(P) of the polynomial P(z). 10 is unsatisfactory for two reasons. First, it applies only to irreducible polynomials.

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Contributions to the Theory of Transcendental Numbers by Gregory V. Chudnovsky
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