Moreover, nit is very glad to have guests from foreign. Nevanlinna theory, also known as value distribution theory, was considered one of the most important areas of research in 193040, so it is not surprising. The purpose of this study is to extend the usual nevan linna theory to a theory for the exact di. Recent development in nevanlinna theory professor min ru university of houston, usa abstract in this talk, we establish a new second main theorem for holomorphic curves in projective space intersecting divisors in general position. Nevanlinnas second main theorem implies that for any meromorphic function f satisfies the defect relation. In 1933 henri cartan proved a fundamental theorem in nevanlinna theory. However, the impact of the new theory was so profound that the process of birth persisted for a period of about ten years. In the beginning we will derive some of these formulae and later use. Jensens formula let us denote by nr,a nfr,a the number of solutions of the equation fz a in the disk z. Rolf nevanlinnas most important mathematical achievement is the value distribution theory of meromorphic functions. In particular, a tropical version of the second main theorem is proved. Mat 205b nevanlinna theory the theory aims to describe the value distribution of meromorphic functions by looking at various formulae connecting the values of meromorphic functions with the distribution of its zeros and poles. Recent development in nevanlinna theory and diophantine approximation3 this is a 1.
Let f z be a nonconstant meromorphic function of zero order, let. A meromorphic function f on c omitting three distinct values of p1 must be constant. This 1929 result of the finnish mathematician rolf nevanlinna 18951980 was described by the distinguished analyst lee rubel as his favourite in all mathematics. Pdf nevanlinna theory in an annulus risto korhonen. We study nevanlinna theory using stochastic calculus. As in diophantine approximation this means that many merely good approximations do exist, which may be the preferable case. We denote by kxm kx p km the relative canonical line bundle. Introduction tropical nevanlinna theory, see 7, describes value distribution of continuous piecewise linear functions. Nevanlinna theory and complex differential equations. The picardborel theorem is a direct consequence of nevanlinna theory. This is especially true of derivatives, which exist in both the function. The roots of the theory go back to the result of emile picard in 1879, showing that a nonconstant complexvalued function which is analytic in the entire complex plane assumes all complex values save at most one. Difference nevanlinna theories with vanishing and infinite periods 3 2 when a. Kunihiko kodaira nevanlinna theory world of digitals.
The exposition is systematic and selfcontained, assuming from the reader some previous background on analytic geometry. In the 1920s, nevanlinna theory was devised by the finnish mathematician rolf nevanlinna, as one of the few great mathematical events of the twentieth century. P1 and approche to a values with the same frequency. The algebraic case i the wellknown fundamental theorem of algebra. It is composed of two main theorems, which are called nevanlinna s. This theory has proved a tool of unparallelled precision for the study of the roots of equations fz a, f1z b, etc.
Hu 31 extended the nevanlinna s rst fundamental theorem to a vectorvalued case with range in hilbert space. Zannier, evertse, and ferretti on combining schmidts subspace theorem with ge ometrical. If we denote by na the number of solutions of the equation fz a, counting multiplicity, in. Nevanlinna theory of the askeywilson divided difference. It has no zeroes or poles, so as a map c p1c it omits the values 0 and theorem picard. The second author and feng have also established a full. Nevanlinna s theory which was published rst in 1925. Nevanlinna theory in several complex variables and. Qualitatively, the first one states that if a value is assumed less frequently than average, then the function comes close to that value more often than average. Keywords schwarzkobayashis lemma hyperbolic measure first main theorem second main theorem bieberbachs example msc 2010.
We present a version of the tropical nevanlinna theory for realvalued, continuous, piecewise linear functions on the real line. It plays an important role in value distribution theory, including generalising picards theorem that an entire function which omits two. Combining this with theorem 7, we have the following theorem. In the philosophy of hasseminkowski principle one hopes to have an arithmetic result if one had have it in padic cases for all prime numbers p, and in the real and complex cases. The aim of this article is to produce another analytical proof of fundamental theorem of algebra, using nevanlinna theory. In particular, we describe the nevanlinna characteristic function tr,f, the order of growth of a meromorphic function, and hadamards factorization theorem. Nevanlinna theory via stochastic calculus sciencedirect. In the 1920s, nevanlinna theory was devised by the finnish math ematician rolf.
Does there exist a meromorphic function fsuch that faj jand fa0for all a62 fajg. For this reason, the theory of value distribution of meromorphic functions concerns itself with questions about the asymptotic behaviour of one term, or, in the invariant sum 1. The nevanlinna theory came into being through the work he did in the years 192224. Nevanlinna theory has since been solidly developed in itself. The book is a valuable treatise on the nevanlinna theory, of special interests to those who want to understand kodairas unique approach to basic questions on complex manifolds. Topics in nevanlinna theory lecture notes in mathematics. The existence or nonexistence of very good approximations is closely related to the diophantine approximation of the first nonzero power series coefficient of at z0. The purpose of this study is to extend the usual nevanlinna theory to a theory for the exact di. The aim of this book is to provide a comprehensive account of higher dimensional nevanlinna theory and its relations with diophantine approximation theory for graduate students and interested researchers.
The notation in the theorem concerning nevanlinna theory will be given in the body of this paper. The purpose of this study is to extend the usual nevanlinna theory to a theory for the exact di erence f 7. Nonarchimedean nevanlinna theory 5045 let fajg be distinct points in p1. Many mathematicians participated in them, who research complex analysis, in particular, complex dynamics theory and the value distribution theory. Nevanlinna s theory in complex function theory, the unit equation 2. It follows from the second fundamental theorem of nevanlinna theory, in turn described by the mighty hermann weyl as one of the greatest achievements of twentieth century mathematics. Nevanlinna theory, diophantine approximation, and numerical. An apoint of a meromorphic function f is said to be cpaired at z. Introduction tropical nevanlinna theory, see 7, describes value distribution of continuous piecewise linear functions of a real variable whose onesided derivatives are integers at every point, similarly as meromorphic. In 1933 henri cartan proved a fundamental theorem in nevanlinna theory, namely a generalization of nevanlinna s second fundamental theorem. In 1980s, ziegler established nevanlinnas theory for the vectorvalued meromorphic functions in finite dimensional spaces. The case 1 above gives an alternative proof of the original little picard theorem. This book with nine chapters systematically describes nevanlinna theory of meromorphic maps between algebraic varieties or complex spaces, building up from the classical theory of meromorphic functions on the complex plane with full proofs in chap.
Unfortunately, it seems that cartans theorem, its proof, and its usefulness, are not as widely known as they deserve to be. Nevanlinna theory in higher dimensions and related. There is no nonconstant holomorphic function c p1c omitting three or more values. Nevanlinna theory and holomorphic mappings between algebraic. This workshop will concentrate on nevanlinna theory value distribution of meromorphic functions and. Nevanlinnas second main theorem is a deep generalisation of the picard theorem. Unlimited viewing of the articlechapter pdf and any. Tropical nevanlinna theory describes value distribution of continuous piecewise linear functions of a real variable with arbitrary real slopes, called tropical meromorphic functions, similarly as meromorphic functions are described in the classical nevanlinna theory. Princeton university press 1943 called it one of the few great mathematical events of the xx century. We have a defect relation for holomorphic maps in equidimensional cases which includes carlson and griffiths.
The results of nevanlinnas theory, which, at the present time, can be regarded as classical, are considered in the rst part of the present book, consisting of chapters i, ii, iii, and sec. As tr does not depend on a, one can say that a meromorphic function takes every value a. I to do so, we simply restrict f to the disc 4r or ar for r suciently large. The purpose of this study is to extend the usual nevanlinna theory to a theory for the exact. Our description of this period will be preceded by a brief summary of the ante nevanlinna value distribution theory. Nevanlinna theory and discuss its applications in the study of the v alue distri bution of zeroorder meromorphic functions. Pdf nevanlinna theory in an annulus, value distribution theory. Zalcmans re scaling method set up a way to combine nevanlinna theory and. Nevanlinna theory nevanlinna theory is part most of value distribution theory of holomorphic functions consider the function ez. System upgrade on tue, may 19th, 2020 at 2am et during this period, ecommerce and registration of new users may not be available for up to 12 hours. Theoretically speaking, we should be able to nd an analogue of any theorem related to the roth theorem in diophantine approximation for nevanlinna theory, and vice versa.
In this paper, we use nevanlinna s second main theorem in the value distribuhow to cite this paper. This book with nine chapters systematically describes nevanlinna theory. Nevanlinna theory provides the most effective tools in the modern theory of meromorphic functions, and even simplifies the older theory of entire functions considerably h. He applied greens formula to prove the first main theorem and the lemma of the logarithmic derivative for meromorphic functions outside a disc of. By combining the first main theorem with the logarithmic deriva tive lemma, we can.
Nevanlinna theory in several complex variables and diophantine approximation. Pdf nevanlinna theory, normal families, and algebraic. Theorem has led to corresponding results in nevanlinna theory. Nevanlinna made the decisive contribution to the development of the theory of value distribution. A concrete presentation of nevanlinna theory in a domain z. The strength of cartans version of nevanlinna theory. Nevanlinna theory is useful in all questions where transcendental meromorphic functions arise, like analytic theory of differential and functional equations holomorphic dynamics, minimal surfaces, and complex hyperbolic geometry, which deals with generalizations of picards theorem to higher dimensions. The main tool is the characteristic function tfr which replaces the degree in the case when f is transcendental. July 20, 2012 value distribution of a rational function f is controlled by its degree d, which is the number of preimages of a generic point. Please, select more pdf files by clicking again on select pdf files. Introduction research institute for mathematical sciences.
After ziegler some works in finite dimensional spaces were done in 1990s 24. Nevanlinna theory outside a disc has been considered in the form of brief. Nevanlinna theory generalizes these facts to transcendental functions f. This book deals with the classical theory of nevanlinna on the value distribution of meromorphic functions of one complex variable, based on minimum prerequisites for complex manifolds. Pdf on may 22, 2017, norbert steinmetz and others published. Nevanlinnas theory provides a new look at some old theorems, such as. His theoremalsoincluded the socalled\rami cationdefects coming. Nevanlinna theory and meromorphic solutions of functional equations. An extension of the nevanlinna theory 191 a premeasure and the associated function is said to be x. Some results of nevanlinna theory in complex banach space e. The classical nevanlinna theory has always attracted many authors and a number of results were developed 9, 10, 17, 29, 25, 31, 32, 40. Nevanlinna theory, or the theory of value distributions, studies the position of the image fa.
Nevanlinna s second theorem shows that, for almost all points, the principal role in the sum 1 is played by. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. The main probabilistic methods used here are some estimates on some increasing processes for brownian motion and martingales on manifolds. Pdf nevanlinna theory for the difference operator researchgate. Diophantine approximation and nevanlinna theory math berkeley. Following a comprehensive treatment of nevanlinnas theory of value distribution, the author presents advances made since haymans work on the value distribution of differential polynomials and illustrates how value and pairsharing problems are linked to algebraic curves and briotbouquet differential equations. Nevanlinna theory is a powerful quantitative tool used to study the growth and behaviour of meromorphic functions on the complex plane. Basicnotations of nevanlinnatheory the theory of meromorphic functions was greatly developed by rolf nevanlinna during the 1920s. By the change of the variable formula or notice that n ft. Cartans theorem works very well for certain kinds of problems. The rolf nevanlinna prize named in honor of rolf nevanlinna is awarded once every 4 years at the international congress of mathematicians, for outstanding contributions in mathematical aspects of information sciences including. To change the order of your pdfs, drag and drop the files as you want. Pdf tropical nevanlinna theory and second main theorem.
Survey on padic nevanlinna theory and its applications to differential equations yang, chungchun. Nevanlinnas value distribution theory or nevanlinna theory is crystallised in its two main theorems. This can be viewed as supporting evidence for the conjecture in ch 1 see also ch 2 that projective algebraic. Hence one would naturally have interest to determine how nevanlinna theory would look in the padic case. On nevanlinna theory for holomorphic curves in abelian varieties yamanoi, katsutoshi, 2004. In 1997, the nevanlinna s rst and second fundamental.