On Nevanlinna's second main theorem in projective space

Ye, Zhuan

doi:10.1007/bf01231453

Cited by 48 publications

(28 citation statements)

References 16 publications

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“…Since then, a number of results have been obtained; see Lang [4], Lang-Cherry [2], Wong [17], Wong-Stoll [18] and Ye [19]. Our proof here basically repeats Ye's argument (cf.…”

Section: Then the Inequalitysupporting

confidence: 75%

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On a General Form of the Second Main Theorem

1997

Trans. Amer. Math. Soc.

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“…Since then, a number of results have been obtained; see Lang [4], Lang-Cherry [2], Wong [17], Wong-Stoll [18] and Ye [19]. Our proof here basically repeats Ye's argument (cf.…”

Section: Then the Inequalitysupporting

confidence: 75%

“…In this paper, we give a proof of a general form of the Second Main Theorem for holomorphic curves in Nevanlinna theory (Theorem 2.3) by using the techniques of Ye [19] to get a good error term. This theorem was first proved by Vojta [15], but with a less precise error term.…”

Section: Introductionmentioning

confidence: 99%

On a General Form of the Second Main Theorem

1997

Trans. Amer. Math. Soc.

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“…It should be noted that those extensions -at least the sronger result in [13] -relied heavily on Nevanlinna's value distribution theory of meromorphic functions, supplemented with some recent developments [22]. The Nevanlinna theory (or, more precisely, Cartan's version thereof) was also the main tool in [11] when proving the appropriate version of Theorem B for entire functions.…”

Section: Introductionmentioning

confidence: 99%

Zeros of analytic functions, with or without multiplicities

Dyakonov

2011

Math. Ann.

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Abstract. The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation a + b = c. It provides a lower bound on the number of distinct zeros of the polynomial abc in terms of deg a, deg b and deg c. We extend this to general analytic functions living on a reasonable bounded domain Ω ⊂ C, rather than on the whole of C. The estimates obtained are sharp, for any Ω, and a generalization of the original result on polynomials can be recovered from them by a limiting argument. IntroductionGiven a polynomial p (in one complex variable), write deg p for the degree of p and N (p) = N C (p) for the number of its distinct zeros in C. The so-called abc theorem, also known as Mason's or Mason-Stothers' theorem, reads as follows.Theorem A. Suppose a, b and c are relatively prime polynomials, not all constants, satisfying a + b = c. ThenOf course, the two sides of (1.1) are positive integers, so (1.1) is equivalent to saying that the left-hand side does not exceed N (abc) − 1.More general variants of this result can be found in Mason's book [15], while the current version is essentially due to Stothers [19]. Various approaches to and consequences of Theorem A are discussed in [10,11,14,17], the most impressive corollary being probably Fermat's Last Theorem for polynomials. The argument leading from abc to Fermat is delightfully simple and elegant, so we take the liberty of reproducing it here.To prove that there are no nontrivial polynomial solutions to the Fermat equation P n + Q n = R n for n ≥ 3, apply the abc theorem with a = P n , b = Q n and c = R n . Write

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“…[7], [15], [20], [4] and [22]) from both fields have started to look for more analogues between these two theories. It is known that a famous theorem of Roth in number theory is analogous to a weak form of the second main theorem in Nevanlinna theory, and the Artin-Whaples product formula in number theory can be viewed as an analogue of the first main theorem in Nevanlinna theory.…”

Section: Introductionmentioning

confidence: 99%

An analogue of continued fractions in number theory for Nevanlinna theory

2004

Trans. Amer. Math. Soc.

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Abstract. We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory. IntroductionSince P. Vojta [17] created a dictionary between Nevanlinna theory and Diophantine approximation theory, researchers (e.g. [7], [15], [20], [4] and [22]) from both fields have started to look for more analogues between these two theories. It is known that a famous theorem of Roth in number theory is analogous to a weak form of the second main theorem in Nevanlinna theory, and the Artin-Whaples product formula in number theory can be viewed as an analogue of the first main theorem in Nevanlinna theory. Theoretically speaking, we should be able to find an analogue of any theorem related to the Roth theorem in Diophantine approximation for Nevanlinna theory, and vice versa. An up-to-date account of these matters appears in [2], [18] and [14]. The author [23] has found an analogue of Khinchin's theorem for Nevanlinna theory, which has given an answer to one of S. Lang's questions in [9]. S. Lang also suggested (in a personal conversation) finding an analogue of the continued fractions for Nevanlinna theory. This is an interesting question that has been around for a while. In this paper, we find an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory. Notation and preliminariesFor the convenience of the general readers, we briefly give some definitions and notation in Nevanlinna theory and continued fractions. Standard references are [3] and [12] for Nevanlinna theory, and [5] and [8] for continued fractions.Let f be a meromorphic function on the whole plane and D r = {|z| < r}. Denote the number of poles of f in D r by n(f, ∞, r), and define n(f, a, r) = n(1/(f − a), ∞, r) if a ∈ C. We also let N (f, a, r) = r 0 n(f, a, t) − n(f, a, 0) t dt + n(f, a, 0) log r.

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On Nevanlinna's second main theorem in projective space

Cited by 48 publications

References 16 publications

On a General Form of the Second Main Theorem

On a General Form of the Second Main Theorem

Zeros of analytic functions, with or without multiplicities

An analogue of continued fractions in number theory for Nevanlinna theory

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