**David Hayes**'s parents were Woodrow R Hayes and Eleanor Crocker. Woodrow Hayes was a musician playing for the Jimmy Poyners Orchestra and as the bandleader of the Woody Hayes Trio. The autobiography [1] is dedicated by Woodrow Hayes to his:-

In addition to their son David, the subject of this biography, Woody and Eleanor Hayes had three daughters: Sally Hayes (now Sally Stevens), Betsy Hayes (now Betsy Hunt), and Jenny Hayes (now Jenny Lewis). David, like his father, attended Needham B Broughton High School in Raleigh, founded in 1929 and one of the oldest of the high schools in Wake County, graduating in the spring of 1955. Later that year he entered Duke University, in Durham, North Carolina, where he majored in mathematics for his Bachelor's Degree which he was awarded in 1959. He remained at Duke University undertaking research for his Ph.D. with Leonard Carlitz as his thesis advisor. Hayes writes [3]:-... wife Eleanor of fifty years, who raised four beautiful children while I was on the road.

Hayes was awarded a Ph.D. by Duke University in 1963 for his 86-page thesisWhile a graduate student at Duke University in the early1960s, I was privileged to take Leonard Carlitz's course "The arithmetic of polynomials." At the first class meeting, Leonard gave each student a detailed set of notes that served as a textbook for the course. These notes were neatly typed and reproduced by ditto machine, the leading-edge copying technique of that era. My own copy, still readable in faded ditto purple, occupies a special place on my office bookshelf. Of course the notes evolved over time. The version I have is277pages long and divided into29chapters.

*The distribution of irreducibles in*GF[

*q*,

*x*]. In it he studied the distribution of irreducibles in arithmetic progressions in a polynomial ring over a finite field. The main results of his thesis were published in a paper in the

*Transactions*of the American Mathematical Society in 1965. This was not his first publication for he had published a 2-page paper

*A polynomial analog of the Goldbach conjecture*in the

*Bulletin*of the American Mathematical Society in 1963. In 1965 he published

*A Goldbach theorem for polynomials with integral coefficients*in the

*American Mathematical Monthly*. D E Littlewood writes in a review:-

After graduating with his Ph.D., Hayes was appointed as an assistant at the University of Tennessee in the autumn of 1963. He worked there for three years before spending the academic year 1966-67 as a Postdoctoral Fellow at Harvard University. He was appointed as an Associate Professor of Mathematics at the University of Massachusetts in September 1967 and, five years later, in the autumn of 1972, he was promoted to a full professorship at the University of Massachusetts. He remained on the faculty there until he retired in 2002. He served as chairman of the Mathematics Department from 1991 to 1994. During his career at Massachusetts, he was the thesis advisor to six Ph.D. students, Michael Nutt (Ph.D. 1974), Gove Effinger (Ph.D. 1981), Mizan Khan (Ph.D. 1990), Daniel Carter (Ph.D. 1993), Zesen Chen (Ph.D. 1996), and Maura Murray (Ph.D. 1999). In addition to his permanent position at Massachusetts, Hayes spent the academic year 1974-75 at the University of Oxford in England. He made research visits to Harvard University in autumn of 1981, to the University of California at San Diego in autumn of 1983, to Imperial College of Science and Technology in London, England, in the spring of 1989, and to Harvard University in autumn of 1999.The author proves a theorem analogous to the Goldbach conjecture that every even integer greater than2can be expressed as the sum of two primes.Z(x)is the domain of polynomials in x with integral coefficients. The theorem is that every polynomial of degree n ≥1can be expressed as a sum of two irreducible polynomials each of degree n.

Michael Rosen writes about some of Hayes other activities [4]:-

Let us look at some of Hayes' publications. In 1966-67 he published three papersFor many years David was a leading figure at the Five College Number Theory Seminar which is held weekly at Amherst College. In the summer of1991, David, along with David Goss and myself, organized a large conference(over100participants from all over the world)at Ohio State University on the subject of "The Arithmetic of Function Fields". The conference proceedings were published a year later by Walter de Gruyter Publishers. David Hayes and I also organized a three day conference "A Mini-Conference on the Arithmetic of Function Fields" held at Brown University in April of1996.

*A polynomial generalized Gauss sum*,

*The expression of a polynomial as a sum of three irreducibles*, and

*A geometric approach to permutation polynomials over a finite field*. A permutation polynomial is one for which the associated polynomial function is a one-to-one mapping of a finite field into itself. L E Dickson classified all permutation polynomials of degree less than seven over a finite field. This classification showed that if the order of the finite field is sufficiently large there are no permutation polynomials of degree 4 or 6. Hayes was able to show that, with certain extra conditions, this result generalised to show that if the finite field is sufficiently large there are no permutation polynomials of even degree. In the second of the three papers he published in 1966-67, Hayes proved a polynomial version of Vinogradov's theorem that every sufficiently large odd integer is the sum of three primes.

Hayes was an invited speaker at many conferences. For example at the 'Number Theory Conference' held at the University of Colorado, Boulder, Colorado in 1972 he delivered the lecture *Adelic analysis in additive number theory*. He:-

At the conference on 'Number theory related to Fermat's last theorem' held at Cambridge, Massachusetts in 1981 he gave the lecture... discussed the Hardy-Littlewood circle method from the point of view of analysis on the ring ofQ-adèles. The singular series associated with a given additive problem is interpreted as a certain Radon-Nikodym derivative.

*Elliptic units in function fields*. Jürgen Ritter writes that, in the lecture, Hayes showed:-

At the conference 'Applications of curves over finite fields' in Seattle, Washington, in 1997 he delivered the lecture... that Robert's results on elliptic units in class fields H of an imaginary quadratic number field F have "more or less" exact analogues over a global function field with finite field of constants. ... The interested reader will certainly appreciate reading the introduction of the paper[in the conference proceedings], in which the general strategy is laid out and which gives much more information ...

*Distribution of minimal ideals in imaginary quadratic function fields*. At the conference 'Stark's conjectures: recent work and new directions' he gave the lecture

*Stickelberger functions for non-abelian Galois extensions of global fields*. In fact the Stark conjectures, both in number fields and in function fields, became the area of mathematics that he worked on most intensely during the last part of his career. We mentioned above that Hayes spent time at Harvard University in the autumn of 1999 and while he was there he delivered a series of lectures on the Stark conjectures. He always intended to use these lectures as the basis for a monograph on the Stark conjectures but he never completed the project. He did, however, publish a monograph in 1991 in collaboration with Gove Effinger (a former Ph.D. student) entitled

*Additive Number Theory of Polynomials Over a Finite Field*.

Hayes' marriage to his first wife ended in divorce but there were three sons from that marriage, Robert Hayes, Christopher Hayes, and Jonathan Hayes. In November 2004 Hayes married Irene (Truchinskas) Brown and became step-father to Carl Dickson, Mark Brown, and James Brown. David and Irene Hayes enjoyed travelling together but sadly they only had six years together before Hayes died on Sunday, 10 April 2011, at the Quabbin Valley Healthcare in Athol. Let us end this biography by quoting the personal comments by Michael Rosen who writes of Hayes' "richness of his interests and the significance of his contributions". Rosen writes [4]:-

I want to make some comments on David's style as a mathematician. He was very serious about his work. He worked on subjects of fundamental mathematical importance and he went deeply into them. He had excellent taste and an abundance of mathematical talent, but also resolve and persistence. If a seemingly insuperable problem appeared, he would keep attempting a solution until all roadblocks were cleared and the way to further progress was open. I deeply admire his work. It is beautiful, important, and of lasting value. Let me conclude with a few personal comments. As time went by our professional relationship turned into a lasting personal friendship. We both liked the outdoors. During my visits to the Amherst area we would often go for hike in the surrounding countryside. My wife and I have a vacation home in New Hampshire. David would visit us there to hike in the autumn and ski in the winter. Of course, he and I would sneak in some mathematical discussions on the side, but mainly these were social occasions where everyone would enjoy the country air, the physical exertions, and the good comradeship. We also enjoyed good food and good wine. We would spend at least part of the evenings indulging in both. Many people who knew David describe him as very intelligent, quiet, modest, and gentle. This is all true. He was also very generous with his time and his ideas. In every way he was a person with exceptional qualities. It was a privilege to have known him and to have been his friend.

**Article by:** *J J O'Connor* and *E F Robertson*