# Charles Pisot

### Quick Info

Born
2 March 1910
Obernai, Alsace-Moselle, Germany (now France)
Died
7 March 1984
Paris, France

Summary
Charles Pisot was a French mathematician who worked in number theory.

### Biography

Charles Pisot's father was the college teacher and historian Othon Pisot who was born on 14 August 1874 at Werentzhouse and died 4 September 1953 at Obernai. He became curator of the museum at Obernai. This town, where Charles was born, is 25 km south west of Strasbourg and, with the rest of Alsace, had been annexed by Germany in 1871 following the Franco-German war of 1870-71. Alsace was returned to France after World War I in 1918 so for the first eight years of his life, Charles was living in Germany. He showed a talent for mathematics at an early age as he explained in the interview [4]:-
I have always been attracted by mathematics. I don't have precise memories but, from what my parents told me, already at the age of two to three years I was particularly interested in numbers, and I amused myself, for example, making calendars ... I still have one that dates from before the war of 1914 and ran until 1952. I knew by heart the correspondences between the days of the month and if it was a Monday, Sunday etc.. for two or three years, around the years 1912, 1913, 1914, ... So I knew how to write numbers before I knew how to write letters. However my father, who was fascinated by the study of history, put me in the literary sections at school. I was not so delighted with that but I got a promise that if I passed with a top A, I would be allowed to study Elementary Mathematics.
Pisot also spoke of other interests he had as a child [4]:-
I have always been attracted by mathematics and by physical things such as astronomy. This fascinated me and when I was a kid I devoured astronomy books more than novels. Novels interested me less than astronomy books but I was interested in physics and also biology.
The school that Pisot attended was in Obernai and he was fortunate that his first teacher of mathematics was an enthusiast for the subject. He explained in [4] how he was taught the method of calculating the square root of numbers and was asked to apply this method to find the square root of 2. He realised that $5^{2} = 25$ and twice 25 is only 1 away from $49 = 7^{2}$. Hence $\large\frac{7}{5}\normalsize$ is a good approximation to √2. He then saw that he could get better and better approximations from the equation $x^{2} - 2y^{2} = 1$. He went to the teacher and showed him what he had discovered and the teacher explained to him that he had discovered the method of continued fractions for himself. The teacher went on to explain more about continued fractions to Pisot who became fascinated - continued fractions would play a large role in his mathematical work from that time on.

Pisot sat the entrance examinations for the École Normale Supérieure on rue d'Ulm and entered this prestigious university in 1929. He was awarded his agrégation in mathematics in 1932 and remained at the École normale supérieure undertaking research. When he explained to the director of the École normale that he wanted to undertake research in number theory, the director told him not to be silly. The director tried to persuade Pisot to undertake research in complex variables but Pisot said he would rather become a school teacher than to be a researcher in complex variables. Of course the director was simply explaining to Pisot that number theory was not an area of interest to French mathematicians at that time [2]:-
French mathematics is filled with the names of leaders in the Theory of Numbers: Fermat, Galois, Lagrange, Liouville and others. Few countries could equal these distinguished names. But curiously, at the time when Charles Pisot began his research career the French contributions to the Theory of Numbers were in an unfortunate eclipse.
However, Pisot did work on number theory for his doctorate, but he had to find his own problems to tackle. His thesis La répartition modulo un et les nombres algébriques was examined on 23 March 1938. Élie Cartan was head of the jury with Paul Montel and Arnaud Denjoy as examiners. Pisot wrote in his thesis:-
I would like to express my deepest thanks to Arnaud Denjoy who has continually encouraged me and helped me throughout this work. I also express sincere thanks to Élie Cartan who has kindly presented the principal results to the Academy of Sciences (4 papers, 2 in 1936 and 2 in 1937), and to Paul Montel who kindly joined Élie Cartan and Arnaud Denjoy to make up the jury. In addition, I thank Leonida Tonelli who has accepted this memoir for the Annals of the Scuola Normale Superiore of Pisa. At the same time I wish to show my gratitude to Claude Chevalley for his help in editing which has been extremely useful to me.
Pisot had published a number of short papers before submitting his thesis: Sur une propriété caractéristique de certains entiers algébraique (1936), Sur certaines propriétés caractéristiques des nombres algébraique (1936), Sur la répartition modulo 1 des puissances successive d'un même nombre (1937), and Sur la répartition modulo 1 (1937). These papers contain his famous 'Pisot numbers', sometimes called the Pisot-Vijayaraghavan numbers', but denoted by $S$ by Pisot himself to honour Raphaël Salem.

The outbreak of World War II dramatically changed to course of Pisot's career. World War II began on 1 September 1939 when German forces entered Poland. On the following day, Britain, France and several other countries, declared war on Germany but over the following months France was not involved in any fighting, but spent time trying to build defences to protect the country from an invasion by Germany. The war changed dramatically for France on 10 May 1940 when the German army crossed the Dutch and Belgium borders and, by June, France had surrendered and fighting had ended. After the fall of France, Pisot was offered a position in a French university but instead he chose to move to Germany. Although this may sound a little strange, one must remember that Pisot was born in Alsace at a time when it was part of Germany. There is no suggestion that he had strong political views but he seems to have preferred to join Germany at this time. Although he had been born in German Alsace, he was still a French citizen and this meant that he could not obtain a permanent university post in Germany without taking German nationality.

Pisot, now known in Germany as Karl Pisot, was given a temporary appointment at the University of Freiburg in 1940. In the following year he was given a temporary position at the University of Greifswald [1]:-
Apparently there were difficulties with Alsations obtaining German citizenship, and without German citizenship Pisot could not possibly get a [permanent] university job in Germany. Efforts seem to have been made on his behalf as early as 1941 or 1942, to no avail. In April 1943 an official at the education ministry wrote that a rectification of Pisot's situation in Germany was called for on grounds of both justice and politics. The official died, and in October 1944 Pisot was still not a German citizen.
Constantin Carathéodory had recommended Pisot for a permanent appointment in 1942 but the citizenship issue prevented this taking place. By the summer of 1944 Pisot was working with Wilhelm Süss at his Research Institute at Freiburg. There he was undertaking military commissions as part of the German war effort. By November 1944 Allied troops were entering Germany and at the end of November the Allies bombed Freiburg. Pisot's family lost everything they possessed in this bombing raid. By this time Wilhelm Süss was working on establishing a National Mathematical Research Institute at Oberwolfach. In early 1945 Süss was arguing strongly for Pisot's German citizenship to be settled so that he could be appointed to the University of Freiburg, the Research Institute at Oberwolfach, or to both. Pisot went with Süss, and other mathematicians from military establishments and colleagues from Freiburg, to Oberwolfach and lived there until the end of the war.

After the war ended, Pisot returned to France and, in 1946, was appointed as an assistant professor at the University of Bordeaux. One might wonder how someone from Alsace who had chosen to support Germany through the war might be accepted in France after hostilities ended [1]:-
While his fellow mathematicians seem to have accepted Pisot, he apparently did suffer recriminations from other French (described in a conversation with Pisot's widow).
In Bordeaux he was asked to teach probability and statistics, topics he had not studied before. He learnt a lot from this experience. In 1948 he was promoted to professor at Bordeaux. He attended the International Congress of Mathematicians in Amsterdam in September 1954 and gave the invited address Sur un ensemble fermé d'entiers algébraique .

He had one doctoral student in Bordeaux but he wanted a larger team of research students. This was a major factor in his decision to move to Paris in 1955 when he was offered a position at the Faculty of Science there. On the one hand he was happy in Bordeaux, but the realisation that he would be able to train many more number theorists in Paris was the major factor in his decision to move. He spoke of his need for students in the interview [4]:-
What is the role of the teacher without students! ... and, they are what you want, when you find a pretty result you must have the audience to explain it to ... and if you do not have students, you cannot tell anyone! ... Mathematics is a passionate adventure and especially when we have students who continue to develop directions we started but we could not continue. It is these students who develop, find some things which are quite new ... I am very happy when that happens.
In addition to working at the Faculty of Science, Pisot also taught at the École Polytechnique. Perhaps one of the best indications of the areas that interested him most is seen from his popular 1960 article Les Nombres entiers, leurs problèmes et leurs mystères, etc . There he looks at diophantine equations, the Goldbach conjecture, Roth's theorem, transcendental numbers, the distribution of primes, and $p$-adic analysis. In Paris, Pisot worked with his colleagues Hubert Delange and Georges Poitou organising the Delange-Pisot-Poitou seminar. This seminar had its origins in the number theory seminar set up by Albert Châtelet in 1947. In collaboration with Mark Zamansky, he published the textbook Mathématiques générales (1958) which proved very popular with students for many years.

In the summer of 1963, Pisot gave a series of lectures at the University of Montreal in Canada. These lectures were published as a book with the title Quelques aspects de la théorie des entiers algébriques . Raphaël Salem, who was a major influence on Pisot, died in the summer of 1963 and Pisot dedicated his book to Salem. D H Lehmer explains in a review that Pisot's lectures:-
... are built around his remarkable theorem to the effect that the set S of all those real algebraic integers q whose conjugates lie inside the unit circle is closed. The author considers wider classes of algebraic numbers that are likewise closed. This is done by applying p-adic analysis to the theory of rational functions and their Taylor expansions. Many other classical and recent results are established as by-products. The ensemble is a very interesting set of theorems.
Sometimes a good conjecture is as important as a good theorem, particularly if it inspires developments as mathematicians attack it. Pisot made the 'Pisot dth root conjecture' which led to considerable progress. This conjecture claimed that if the coefficients $b(n)$ of the Taylor expansion of a rational function all have perfect $d$ th roots for $n > N$, then one can choose a $d$ th root $a(n)$ of $b(n)$ for each $n$ such that $Sa(n)x^{n}$ is a rational function. Various special cases of the conjecture were proved true before Umberto Zannier gave a complete proof of the conjecture in the year 2000 (see [7]).

Of course, French mathematics was for many years dominated by the Bourbaki group of mathematicians and Pisot was invited to join them. However, he found that the theory of numbers did not fit into the overall scheme of mathematics as envisioned by Bourbaki. He said in the interview [4]:-
A number of mathematicians created Bourbaki to try to introduce structures into mathematics and I was asked to participate. I was given the task of trying to find structures in the theory of numbers, but this does not work, there is no structure there and finally Bourbaki gave up trying to do something in the theory of numbers. Now I begin to wonder why: I think that mathematics on the whole stems from two sources. The first source, which obviously everyone thinks of, is experiments, and physics and chemistry ... . These areas pose some problems that advance mathematics, but there is one source that seems equally important which is the theory of numbers. The problems posed by the integers require such work and such reflections that ultimately it is from there that almost half of mathematical theories arise.
In [5] Pisot relates his own research experiences with the aim of being helpful to young researchers as he states in that article:-
Having taught for fifty years, I have seen some young, initially enthusiastic researchers become disappointed by their efforts, either because they were stumped by the difficulty of the question examined or because they realized they had re-established known results. I thought therefore that an account of my own development might perhaps keep some of them from getting discouraged.
As we have explained, students were always of major importance to Pisot. The article [2] gives this example of his influence through his students:-
The number of students of Charles Pisot, from all parts of France and also some from foreign countries, is of the order of twenty. But this underestimates the influence of Charles Pisot on the Theory of Numbers by only considering those who were directly his pupils. Patrice Philippon, who submitted his thesis in 1984, was a pupil of Daniel Bertrand (submitted 1977), who was a pupil of Michel Waldschmidt (submitted 1972), who was a pupil of Jean Fresnel (submitted 1967), who was a pupil of Yvette Amice (submitted 1965), who was herself a pupil of Charles Pisot. That is one example, but there are many others ...
In fact the chain described in this quote continues with Marc Chardin (submitted 1990) being a student of Patrice Philippon and Seyed Hamid Hassanzadeh (submitted 2009) being a student of Marc Chardin.

Pisot retired from his positions in Paris in 1979. He had received several honours for his outstanding contributions including the Dickson Prize from the Académie des Sciences in 1947 and the city of Paris prize, also awarded by the Académie des Sciences, in 1966.

### References (show)

1. S L Segal, Mathematicians under the Nazis (Princeton University Press, Princeton, 2003).
2. Y Amice, M-J Bertin, F Bertrandias, A Decomps, F Dress, M Grandet, M Mendès France and G Rauzy, Charles Pisot, Acta Arithmetica 51 (1) (1988), 1-4.
3. J-P Kahane, Sur une note de Charles Pisot, Acta Arithmetica 34 (1979), 267-272.
4. J Nimier, Entretien avec le Mathematicien Charles Pisot. http://www.pedagopsy.eu/entretien_pisot.htm
5. C Pisot, Souvenirs mathématiques, Enseign. Math. (2) 26 (3-4) (1980), 185-192.
6. Publications of Charles Pisot, Acta Arithmetica 51 (1) (1988), 5-7.
7. U Zannier, A Proof of Pisot's dth Root Conjecture, Ann. Math. 151 (1) (2000), 375-383.