## Functional Equations in Arithmetic

### Cincinnati, Ohio 1982

**JULIA BOWMAN ROBINSON** was born in St. Louis, Missouri in 1919. She began college majoring in mathematics, in order to receive public school teaching credentials, but later transferred to the University of California at Berkeley as her interest shifted to research mathematics. She received her BA in 1940 and began her graduate studies after discovering that potential employers were more interested in her typing skills than her mathematics.

At Berkeley, she studied number theory with Raphael M. Robinson. They married in 1941, after which nepotism rules prohibited her from teaching as a graduate assistant in Berkeley's mathematics department. In 1947, she began work with the logician Alfred Tarski for her doctorate, which she received in 1948. Her thesis showed that the notion of an integer can be defined arithmetically in terms of the notion of a rational number and the operations of addition and multiplication on the rationals. The arithmetic of rationals is therefore adequate for the formulation of all problems of elementary number theory.

In 1948 Robinson began work on the tenth problem on Hilbert's famous list: "to find an effective method for determining if a given diophantine equation is solvable in integers." Although she also published papers on a variety of questions, the tenth problem was to occupy the largest portion of her professional career. Her early results took on added importance in 1961 with the publication of a joint paper with Martin Davis and Hilary Putnam in which it was proved that every recursively enumerable set is existentially definable in terms of exponentiation and that, therefore, there is no algorithm for deciding whether an exponential diophantine equation has a solution in natural numbers. In view of her earlier proof that exponentiation is existentially definable in terms of any function of roughly exponential growth, the negative solution of Hilbert's problem was reduced to finding an existential definition of such a function. In 1970, a young mathematician in Leningrad, Yuri Matuasevic, completed the proof.

In 1975, Robinson became the first woman mathematician to be elected to the National Academy of Sciences, and, in 1983, she became the first woman president of the American Mathematical Society. Her other honors included election to the American Academy of Arts and Sciences, a grant from the MacArthur Foundation, and an honorary degree from Smith College. She died in 1985.

After her death, the *Notices of the AMS* (November 1985, pages 739-42) published a number of tributes written by Robinson's colleagues. Elizabeth Scott, who knew Robinson from their days in graduate school together, described some of Robinson's difficulties in securing a position at Berkeley. At one point, writes Scott, Robinson was required to submit a description of what she did each day to Berkeley's personnel office. So she did: "Monday--tried to prove theorem, Tuesday--tried to prove theorem, Wednesday--tried to prove theorem, Thursday--tried to prove theorem; Friday--theorem false." The personnel office then let the graduate division handle Robinson's appointment. "Throughout her life Julia stood up for offering opportunities to all students," wrote Scott. "She also encouraged graduate students and young faculty to have more confidence in their real abilities. She felt that women and minority mathematicians especially needed this support, which she provided with spirit yet in a quiet way... She encouraged us to work together so that all women who have the ability and the desire to do mathematical research can have the opportunity to do so."