Theatre B, Richard Berry Building
The University of Melbourne
Centre for Number Theory Research
Macquarie University, Sydney
Our forebears knew that one can integrate rational functions, in some variable and in the square root of a polynomial of degree two in that variable, in terms of logarithms --- recall that inverse trigonometric and hyperbolic functions are just logarithms in slight disguise. But when it comes to the case where the polynomial has degree greater than two, nasty things happen: one obtains inverse elliptic functions and worse. Nonetheless, several centuries ago, Abel noticed that certain integrals which morally should be expected to be nasty indeed were not, refusing to deliver more than an innocent logarithm of an algebraic function. Abel went on to study, well, abelian integrals, and it is Chebychev who explains --- using continued fractions --- what is going on with these `quasi-elliptic' integrals. Recently, a student of mine, Xuan Chuong Tran, computed all the polynomials of degree four which permit the phenomenon to occur. I will explain various contexts in which the exceptional polynomials (better, the exceptional algebraic curves) arise. They include symbolic integration of algebraic functions; the study of units in function fields; diophantine approximation of algebraic numbers; and, given a polynomial, the consideration of period length of the continued fraction expansion of the square root of that polynomial evaluated on the sequence of integers. The underlying secret turns out to be the behaviour of the continued fraction expansion of the square root of a polynomial. So the core of my remarks will be the story of periodic continued fraction expansions both in number fields and in function fields. Readers should pause to admire the felicitous manner in which I have carefully deTeXed this abstract (not by choice, but under the whip of the web). Once it had many $$s, now it is worthless.