The second main theorem for small functions and related problems

By K. Yamanoi. 2004

209 citations recorded by the source corpus.

We shall establish the following three results in more general forms. (1) The second main theorem for small functions. Let f be a meromorphic function on the complex plane C. Let a1, . . . , aq be distinct meromorphic functions on C. Assume that ai are small with respect to f ; i.e., T (r, a) 0 (Corollary 1). Here as usual in Nevanlinna theory, the terms T (r, f) and N(r, ai, f) denote for the characteristic function and the truncated counting function, respectively. (2) Application to functional equations. Let KC be the field of meromorphic functions on C. For a function ψ : R>0 → R, put KψC = {a ∈ KC; T (r, a) ≤ O(ψ(r)) ||}, which is a subfield of KC. Then the following holds: Let F (x, y) ∈ KψC [x, y] be a polynomial in two variables over K ψ C . Assume that the curve F (x, y) = 0 over KψC has genus greater than one. If ζ1, ζ2 ∈ KC satisfy the functional equation F (ζ1, ζ2) = 0, then both ζ1 and ζ2 are contained in KψC (Corollary 2). (3) Height inequality for curves over function fields. Let k be a function field of one variable over C. Let X be a smooth projective curve over k, let D ⊂ X be a reduced divisor, let L be an ample line bundle on X and let e > 0. Then we have hk,KX (D)(P ) ≤ N (1) k,S(D,P ) + dk(P ) + ehk,L(P ) +Oe(1) for all P ∈ X(k)\D (Theorem 5). Here the notations are introduced in [V1], [V3] (see also section 9). Our proof uses Ahlfors’ theory of covering surfaces and the geometry of the moduli space of q-pointed stable curves of genus 0.

Subfields: Applied Mathematics

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In Applied Mathematics

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2. Ting-Bin Cao
3. Ling Xu
4. I. Laine
5. R. Halburd
6. J. Heittokangas
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Citations in the corpus

• Nevanlinna theory for the difference operator (2005, background): “A complete solution to this problem was given recently by Yamanoi [14] who showed that (1.1) indeed remains valid if the sum is taken over the larger field small functions, rather than just constants.”
• ABC implies primitive prime divisors in arithmetic dynamics (2012, methodology): “Theorem 4.1 (Yamanoi).”
• Uniqueness of meromorphic functions sharing values with their shifts (2011, background): “Combining (10) and (11) with the second main theorem [21], it follows that”
• Existence of finite-order meromorphic solutions as a detector of integrability in difference equations (2006, unknown): “, [17–19]).”
• Logarithmic difference lemma in several complex variables and partial difference equations (2018, background): “1] that the conclusion is easily extended from the second main theorem for small function targets due to Yamanoi [46] by the standard process of averaging over the complex lines in C.”