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On Some Applications of Diophantine Approximations

Umberto Zannier, editor
Edizioni della Normale
Publication Date: 
Number of Pages: 
Publications of the Scuola Normale Superiore 2
[Reviewed by
Fernando Q. Gouvêa
, on

It used to be easy to make sure that everyone who mattered could read your book or paper: just write it in Latin. This began to change in France in the 18th century and in Germany in the 19th. Slowly, people began to write in their native language, and scholars began to have to learn those languages. At many universities, language requirements became part of Ph.D. programs: serious scholars had to be able to read the languages that were important in their discipline. When I was a student, the choices for a future mathematician were French, German, and Russian, all of which were languages in which there was a substantial mathematical literature.

In recent decades, this has changed a little, as more and more scholars have started to write in English. It’s simply good practice: you want people to be able to read your papers, after all. In response, many graduate schools seem to have given up on language requirements. Historians of our period will have to learn English, just as historians of early modern mathematics have to learn Latin.

Between 1850 and 1950, however, quite a lot of very important mathematics (especially in number theory) was written in German, from the Dirichlet-Dedekind Zahlentheorie to the work of Noether and Artin, Hecke and Hasse. People interested in that period must either learn German or rely on translations. Those in the latter camp will want a copy of this book.

Carl Ludwig Siegel was brilliant, as we all know, and he wrote many fundamental papers. One of them, published in 1929, was called “Über einige Anwendungen diophantischer Approximationen,” which translates to On Some Applications of Diophantine Approximations; this book contains both Siegel’s original paper in German and an English translation. The translator and editor have added a few notes, an introduction, and a brief survey of the impact of Siegel’s paper.

The paper itself has two parts. In the first, Siegel sharpens the results on Diophantine approximation known at his time and uses them to obtain several theorems about transcendental numbers. (This was before Roth’s theorem, which means Siegel has to work harder than we would today.) In the second, he uses his results to prove a very famous theorem about integral points on (affine models of) algebraic curves. So it is clear that this is a fundamental paper, well worth reading.

The translation is serviceable, though the English is sometimes a little German. The notes are very useful, and the editors have been careful to mark each of them as “Footnote by the Editors” (even though the original paper has no footnotes). Edizioni della Normale has done a beautiful job of producing the book. It will be of great interest to mathematicians working in transcendence theory and Diophantine approximation, and to anyone interested in the history of mathematics in the early 20th century.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He can almost read German.

See the table of contents on the publisher's web page for the book.