## Wolfgang M Schmidt's Books

Wolfgang M Schmidt has published a number of influential books. We list eight of these books below and give a little further information such as extracts from reviews and extracts from Prefaces.

**1. Approximation to algebraic numbers (1972), by Wolfgang M Schmidt.**

**1.1. Note.**

Survey lectures given at the Institute for Advanced Study, Princeton, under the sponsorship of the International Mathematical Union.

**1.2. Chapter headings.**

Introduction; An outline of the proof of Roth's Theorem; Some generalizations of Roth's Theorem; Effective methods Baker's Theorem; Simultaneous approximation to algebraic numbers by rationals; Tools from the Geometry of Numbers; Norm forms.

**1.3. Review by: Alan Baker.**

*Mathematical Reviews*MR0327672

**(48 #6014)**.

This is an excellent survey article describing some of the extensive ground that has been won in recent years in the field of Diophantine approximation. The first half includes a resumé of the classical results of Dirichlet, Hurwitz, Hin in and others, a discussion of the famous theorem of Roth on the approximation of algebraic numbers by rationals, and a description of the reviewer's effective methods in this context. The second half is devoted to the author's fundamental generalization of Roth's theorem on the simultaneous approximation of algebraic numbers by rationals. A clear account is given of the main new ideas involved in the proof, such as the construction of the auxiliary polynomial in many variables homogeneous in blocks, and the use of the geometry of numbers to show that propositions on the penultimate minimum of a certain parallelepiped imply similar propositions for the first minimum. All significant applications of the principal theorems are recorded in detail, and there is also a useful list of references to related work.

**2. Equations over finite fields: an elementary approach (1976), by Wolfgang M Schmidt.**

**2.1. From the Preface.**

These Lecture Notes were prepared from notes taken by M Ratliff and K Spackman of lectures given at the University of Colorado. I have tried to present a proof as simple as possible of Weil's theorem on curves over finite fields. The notions of "simple" or "elementary" have different interpretations, but I believe that for a reader who is unfamiliar with algebraic geometry, perhaps even with algebraic functions in one variable, the simplest method is the one which originated with Stepanov. Hence it is this method that I follow. The length of these Notes is perhaps shocking. However, it should be noted that only Chapter I and III deal with Weil's theorem. Furthermore, the style is (I believe) leisurely, and several results are proved in more than one way. I start in Chapter I with the simplest case, i.e., with curves y

^{d}= f(x). At first I do the simplest subcase, i.e., the case when the field is the prime field and when d is coprime to the degree of f. This special case is now so easy that it could be presented to undergraduates. The general equation f(x, y) = 0 is taken up only in Chapter III, but a reader in a hurry could start there. The second chapter, on character sums and exponential sums, is included at such an early stage because of the many applications in number theory. Chapters IV, V and VI deal with equations in an arbitrary number of variables. Possible sequences are chapters 1 by itself, or 1, 3 for Weil's theorem, or 1.1, 3 for a reader who is in a hurry, or 1, 2 for character sums and exponential sums, or 1, 2, 4, or 1, 3, 4.3 and 5. Originally I had planned to include Bombieri's version of the Stepanov method. I did include it in my lectures at the University of Colorado, but I first had to prove the Riemann-Roch Theorem and basic properties of the zeta function of a curve. A proof of these basic Properties in the Lecture Notes would have made these unduly long, while their omission would have made the Bombieri version not self complete. Hence I decided after some hesitation to exclude this version from the Notes . Recently Deligne proved far reaching generalizations of Weil's theorem to non-singular equations in several variables, thereby confirming conjectures of Weil. It is to be noted, however, that Deligne's proof rests on an assertion of Grothendieck concerning a certain fixed point theorem. To the best of my knowledge, a proof of this fixed point theorem has not appeared in print yet. It is perhaps needless to say that at present there is no elementary approach to such a generalization of Weil's theorem. But it is to be hoped that some day such an approach will become available, at least for those cases which are used most often in analytic number theory.

November, 1975

W M Schmidt

**2.2. Review by: H M Stark.**

*Mathematical Reviews*MR0429733

**(55 #2744)**.

Recently in a series of papers beginning in 1969 S A Stepanov introduced a new method in number theory for estimating the number of solutions of equations over finite fields. Using this method, the author [Wolfgang M Schmidt] and E Bombieri independently gave proofs of the Riemann hypothesis for curves. Thus, it is now possible to give a self-contained treatment for number theorists and graduate students of results which formerly required a large background in algebraic geometry. This is the purpose of the present set of lecture notes which are based on part of a course given by the author. That course included the Riemann-Roch theorem and Bombieri's proof but the author has unfortunately deleted this material for fear that the notes would be too long.

**3. Small fractional parts of polynomials (1977), by Wolfgang M Schmidt.**

**3.1. From the Preface.**

Our knowledge about fractional parts of linear polynomials is fairly satisfactory. ... Our knowledge about fractional parts of nonlinear polynomials is not so satisfactory. In these Notes we start out with Heilbronn's Theorem on quadratic polynomials ... from this we branch out in three directions. In ¤¤7-12 we deal with arbitrary polynomials with constant term zero. In ¤¤13-19 we take up simultaneous approximation of quadratic polynomials, and in ¤¤20, 21 we discuss special quadratic polynomials in several variables. There are many open questions; in fact, most of the results obtained in these Notes are almost certainly not best possible. Since the theory is not in its final form, I have refrained from including the most general situation, i.e. simultaneous fractional parts of polynomials in several variables of arbitrary degree. On the other hand, I have given all the proofs in full detail, at a leisurely pace.

**3.2. Review by: Ming-Chit Liu.**

*Mathematical Reviews*MR0457360

**(56 #15568)**.

In this book the author provides a comprehensive account of the recent major discoveries in generalizations of Heilbronn's result [of 1948 which generalises Dirchlet's theorem]. Among them many are significant improvements in both the methods and results due to the author himself. The book consists of twenty-one sections. Heilbronn's theorem is introduced in Section 1 together with three open problems. ... The author's style is elegant and the material is clearly throughout. With only a few exceptions the author provides all proofs in full detail even for Vinogradov's lemma and for Weyl's inequality. The book is readable and contains many useful remarks and some interesting conjectures which will certainly interest both specialists and nonspecialists.

**4. Lectures on irregularities of distribution (1977), by Wolfgang M Schmidt.**

**4.1. From the Preface.**

These lectures were given at the Tata Institute of Fundamental Research, Bombay, in the Fall of 1972. Excellent notes were taken by T N Shorey. The theory of Irregularities of Distribution began as a branch of Uniform Distributions, but is of independent interest. The papers appearing in 1922 of Hardy and Littlewood and of Ostrowski on fractional parts of sequences may be regarded as forerunners of the general theory. The first papers dealing with the distribution of general sequences are due to T Van Aardenne-Ehrenfest in 1945, 1949, and K F Roth in 1954.

**4.2. Review by: Henri Faure.**

*Mathematical Reviews*MR0554923

**(81d:10047)**.

These lectures were given in 1972, but the writing of the notes was updated in 1976. The course is divided into two parts; the first presents the fundamental results on the discreteness of the sequences in the k-dimensional torus ..., and ends with a study of bounded and unbounded residual intervals. ... This course, built around the first nine issues of the author's "Irregularities of distribution" series, is oriented towards distribution problems with a geometric interpretation.

**5. Diophantine approximation (1980), by Wolfgang M Schmidt.**

**5.1. From the Preface.**

In Spring 1970 I gave a course in Diophantine Approximation at the University of Colorado, which culminated in simultaneous approximation to algebraic numbers. A limited supply of mimeographed Lecture Notes was soon gone. The completion of these new Notes was greatly delayed by my decision to add further material. The present chapter on simultaneous approximations to algebraic numbers is much more general than the one in the original Notes. This generality is necessary to supply a basis for the subsequent chapter on norm form equations. There is a new last chapter on approximation by algebraic numbers. I wish to thank all those, in particular Professor C L Siegel, who have pointed out a number of mistakes in the original Notes. I hope that not too many new mistakes have crept into these new Notes. The present Notes contain only a small part of the theory of Diophantine Approximation. The main emphasis is on approximation to algebraic numbers. But even here not everything is included. I follow the approach which was initiated by Thue in 1908, and further developed by Siegel and by Roth, but I do not include the effective results due to Baker. Not included is approximation in p-adic fields, for which see e.g. Schlickewei [Die p-adische Verallgemeinerung des Satzes von Thue-Siegel-Roth-Schmidt (1976)].

**5.2. Review by: A J van der Poorten.**

*Mathematical Reviews*MR0568710

**(81j:10038)**.

The present book is based on mimeographed notes ["Lectures on Diophantine approximation", 1970] as well as on an article of the author [Enseign. Math. (2) 17 (1971)]. The original notes have been augmented by considerably more material on simultaneous approximations to algebraic numbers so as to provide a basis for the subsequent chapter on norm form equations. There is a new last chapter on approximation by algebraic numbers. These notes will provide an invaluable aid both for the student and for the specialist. Indeed much of the material is not available other than in technical papers, and the relatively gentle pace of presentation makes the material accessible (though not easy). Of course these notes by no means cover the entire field of Diophantine approximation but the topics selected form a coherent whole, and are dealt with to considerable depth and useful detail.

**6. Analytische Methoden für Diophantische Gleichungen. Einführende Vorlesungen (1984), by Wolfgang M Schmidt.**

**6.1. Review by: D J Lewis.**

*Mathematical Reviews*MR0772928

**(86f:11024)**.

This monograph is based on lectures of the author in Vienna and Düsseldorf in 1983 and provides a careful introduction to the use of the Hardy-Littlewood method in the study of rational points on homogeneous varieties including the author's most recent research in this field. ... [it] will undoubtedly be the basic reference for this subject for some time to come.

**7. Diophantine approximations and Diophantine equations (1991), by Wolfgang M Schmidt.**

**7.1. From the Preface.**

The present notes are the outcome of lectures I gave at Columbia University in the fall of 1987, and at the University of Colorado 1988/1989. Although there is necessarily some overlap with my earlier Lecture Notes on Diophantine Approximation (Springer Lecture Notes 785, 1980), this overlap is small. In general, whereas in the earlier Notes I gave a systematic exposition with all the proofs, the present notes present a variety of topics, and sometimes quote from the literature without giving proofs. Nevertheless, I believe that the pace is again leisurely. Chapter I contains a fairly thorough discussion of Siegel's Lemma and of heights. Chapter II is devoted to Roth's Theorem. Rather than Roth's Lemma, I use a generalization of Dyson's Lemma as given by Esnault and Viehweg. A proof of this generalized lemma is not given; it is beyond the scope of the present notes. An advantage of the lemma is that it leads to new bounds on the number of exceptional approximations in Roth's Theorem, as given recently by Bombieri and Van der Poorten. These bounds turn out to be best possible in some sense. Chapter III deals with the Thue equation. Among the recent developments are bounds by Bombieri and author on the number of solutions of such equations, and by Mueller and the author on the number of solutions or Thue equations with few nonzero coefficients, say s such coefficients (apart from the constant term). I give a proof of the former, but deal with the latter only up to s = 3, i.e., to trinomial Thue equations. Chapter IV is about S-unit equations and hyperelliptic equations. S-unit equations include equations such as 2

^{x}+ 3

^{y}= 4

^{z.}I present Evertse's remarkable bounds for such equations. As for elliptic and hyperelliptic equations, I mention a few basic facts, often without proofs, and proceed to counting the number of solutions as in recent works or Evertse, and of Silverman, where the connection with the Mordell-Weil rank is explored. Chapter V is on certain diophantine equations in more than two variables. A tool here is my Subspace Theorem, of which I quote several versions, but without proofs. I study generalized S-unit equations, ... as well as norm form equations. Recent advances permit to give explicit estimates on the number of solutions. The notes end with an Epilogue on the abc-conjecture of Oesterlé and Massr.

**7.2. From an Acta Scientiarum Mathematicarum review.**

This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers.

**7.3. From an L'Enseignement Mathematique Review.**

The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well.

**7.4. Review by: Carlo Viola.**

*Mathematical Reviews*MR1176315

**(94f:11059)**.

This book arises from a series of lectures given by the author at Columbia University in the autumn of 1987, which the reviewer had the pleasure of attending, and at the University of Colorado in 1988/89, though some later results are also included. To some extent the book can be considered as a continuation of the author's earlier lecture notes on Diophantine approximation [Diophantine approximation, 1980], since the overlap with that book is small. Unlike its predecessor, however, the present book is not intended to give all the proofs of the results included, partly because the omitted proofs of some recent theorems, which would require highly technical tools from various fields such as algebraic or Diophantine geometry, would considerably increase the length and perhaps harm the brilliant style and the gentle pace of the exposition. Naturally, all the original papers are quoted, and the text is enriched with an ample bibliography. As is plain from the table of contents, this book is not meant as an exhaustive description of the state of the art in Diophantine approximation and equations, but is rather a coherent selection of topics culminating in some recent achievements concerning bounds on the number of solutions of various Diophantine equations.

**8. Equations over finite fields: an elementary approach (2nd edition) (2004), by Wolfgang M Schmidt.**

**8.1. From the Preface.**

Despite some new developments, I decided to leave the original presentation unchanged. But note that the Grothendieck assertion was proved around the time when the original manuscript was prepared, thus establishing the truth of Deligne's theorem.

**8.2. Review by: Yann Bugeaud.**

*Mathematical Reviews*MR2121285

**(2005k:11124)**.

In 1948, A Weil published the proof of the Riemann Hypothesis for function fields in one variable over a finite ground field. New, elementary proofs of Weil's most significant results were given around twenty years later by S Stepanov, using ideas from Diophantine approximation. Subsequently, W M Schmidt and E Bombieri followed Stepanov's method to get proofs of Weil's result in full generality. The book under review contains accounts of both methods. Schmidt's method, which is more elementary, is discussed in the first six chapters, together with many related matters. The remaining three chapters cover Bombieri's proof. The book under review was prepared in 1975 from notes of lectures given by the author. Despite some new developments, he chose to leave the original presentation unchanged. ... No substantial prerequisites are needed to read the present book. It is written in a very leisurely style, contains many examples, and the proofs are given in full detail. It is a most valuable reference for graduate students and researchers.