Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better
[A short version of this paper has been published in Contemp. Math. 25, 129-130 (1983; Zbl 0528.30020).] This paper proves two results by a common construction. The first result is in Nevanlinna Theory. A fundamental result due to Picard says that a nonconstant meromorphic function assumes every value with at most two exceptions. In the 20th century Nevanlinna showed that there is an asymptotic bound on the number of solutions of \(f(z)=a\) in circles centered about \(z=0\), with equality holding for all but an at most countable set of a's. For those values of a for which this asymptotic bound is not attained he defined a number called the defect that can range between 0 and 1. Nevanlinna showed that the sum of all of the defects of a nonconstant meromorphic function is at most 2. Nevanlinna conjectured but could not show that if the set of possible a values was enlarged to include very small growth meromorphic functions the bound of 2 would still hold. In this paper the author gives the first proof of Nevanlinna's conjecture with no unnatural restrictions on the large meromorphic function. The paper is lengthy in part because of the method of proof and in part because of the number of results that it establishes in both complex variables and number theory. His Theorem III is a generalization of the n small function theorem to linear forms. That it really is a generalization is established with equation (21). The number-theoretic parts of this paper deal with obtaining effectively computable lower bounds on the approximation of certain types of formal power series. A typical result bounds the goodness of the approximation of formal power series solutions to linear differential equations having polynomial coefficients, by rational functions. The measure of goodness of such approximations is taken to be the number of places of exact agreement between the two functions at a set of chosen points in the extended plane. Here the rational function plays the part of the large meromorphic function. More generally, lower bounds were obtained on the closeness to the zero function of linear combinations of such functions having polynomial coefficients (using the same measure of closeness). The author, as he acknowledges, was narrowly anticipated in submission for publication by \textit{D. V. and G. V. Chudnovsky} [Proc. Natl. Acad. Sci. USA 80, 5158-5162 (1983; Zbl 0516.12019)] in the cases of several of the number-theoretic results. See the note in proof on p. 388, for details. Effective bounds on the approximation of values of Siegel E- and G-functions can be established using the number-theoretic results. In a paper to appear in Monatsh. Math. in the Fall of 1986, the author establishes such bounds. They appear to be somewhat stronger than similar results obtained by the Chudnovskys. Beginning with the author's earlier work on approximating power series it becomes clear one way of obtaining such lower bounds was to construct an algebraic differential equation that is satisfied by each formal power series to be approximated and that also fulfills a collection of other technical conditions, the most difficult of these is that the approximating rational functions should not satisfy this differential equation. This method was used by the author in proving his results, by the Chudnovskys in their work on the number-theoretic case, and by Norbert Steinmetz in a simplified proof of the n-small function theorem that has recently been widely communicated informally. What remains to be seen is how the author constructed the requisite differential equation. Being a number-theorist, he used the proof of the Thue-\(Siegel\)-\(Roth\)-\(Schmidt\) Theorem as a guide. One formulation of the Schmidt version of this theorem gives lower bounds on the absolute values of products of m fixed linearly independent linear forms, in m integral value parameters, each having linearly independent algebraic numbers as coefficients, while the variables, considered as forming a vector \(\bar v,\) range over nonzero vectors of integers. Speaking roughly, this bound is given in terms of the absolute value of the product of the coefficients. The proof of the Schmidt Theorem is carried out by constructing polynomials in Nm variables, that do not vanish at an Nm dimensional vector of integers \(\bar w\) whose entries are those occurring in a sequence of N of the \(\bar v,\) for some large positive integer N. Further it is assumed that each of these \(\bar v\) corresponds to a set of m linear forms having quite small product (so small that by the Schmidt theorem itself there can be only finitely many of them). A proof by contradiction is obtained; the contradiction is that too many such sets of m linear forms having quite small product were assumed to exist. The author's idea was to have the part of the different \(\bar v,\) now \(\bar v(\)z), be played by one \(\bar v(\)z), in effect, cloning it by successively replacing z with \(z_ 1,...,z_ N\) in the entries of \(\bar v(\)z) and use these Nm entries to form his \(\bar w.\) He shows that a polynomial can be constructed in direct analogy with the proof of Schmidt's theorem that does not vanish as a function of \(z_ 1,z_ 2,...,z_ N\) when evaluated at the entries of \(\bar w.\) One at first wishes to set \(z=z_ 1=z_ 2=...=z_ N\) and obtain a nonzero polynomial in z. That would be too much to expect; what does work is finding a partial derivative of the expression in \(z_ 1,z_ 2,...,z_ N\) that does not vanish at the point \(z=z_ 1=z_ 2=...=z_ N\). The existence of such a partial is guaranteed by the nonvanishing of a certain Wronskian. It is possible to use this nonvanishing Wronskian directly, constructing the requisite differential equation directly without reference to the Thue-\(Siegel\)-\(Roth\)-\(Schmidt\) Theorem, as first the Chudnovskys and then N. Steinmetz noted. The same author in a paper that should be better known several years ago showed that a meromorphic function can not be totally ramified at 5 small algebroid points [the author and the reviewer, Lect. Notes Pure Appl. Math. 78, 25-33 (1982; Zbl 0488.30016)]. A few corrections: on p. 354, line 21, \(\delta^ c_{k_ 1(u)}\) should be \(\delta^ v_{k_ 1(u)}\); on p. 359, line 3, \(f_{11}\) should be \(f_{111}\); on p. 363, line 4, \(1\leq M\leq q\) should be \(1\leq m\leq q\), and in lines 9 and 11, \(A_{ijkm}\) should be \(A_{ikm}\); on p. 372, line 3, \(u_ 1,u_ 2^{-1}\) should be \(u_ 1u_ 2^{-1}\); on p. 379, the third line from the bottom, \(v^{-1}_{nk}k(j)\) should be \(v^{-1}_{nk(j)}\); and on page 387, line 13, hupothesized should be hypothesized.
generalization of the n small function theorem
solutions to linear differential equations
Thue-\(Siegel\)-\(Roth\)-\(Schmidt\) Theorem
scientific article; zbMATH DE number 3918476
scientific article