Joint UBC-SFU Number Theory Seminar

Thursday February 21, 2002
in K9509, SFU

3:00 - 3:45
Dr Alf van der Poorten
Centre for Number Theory Research, Macquarie University, Sydney, Australia

Title: "Pseudo-elliptic Integrals"

Abstract: Several centuries ago, Abel noticed that the well-known elementary integral $$\int \frac{dx}{\sqrt{x^2+2bx+c}}=\log(x+b+\sqrt{x^2+2bx+c})$$ is an augur of rather more unexpected integrals of the shape $$\int \frac{f(x)dx}{\sqrt{D(x)}}=\log(p(x)+q(x)\sqrt{D(x)}).$$ Here the $D$ are certain polynomials of degree $\deg D(x)=2g+2$ and the $f$ are polynomials of degree just $\deg f(x)=g$. Specifically, $f(x)=p'(x)/q(x)$ (so $q$ divides the derivative $p'$ of $p$). An example is \begin{multline*} \int\frac{6x\,dx}{\sqrt{x^4+4x^3-6x^2+4x+1}}= \log\Bigl(x^6 + 12x^5 + {45}x^4+ 44x^3 - {33}x^2 + 43\\+ (x^4 + 10x^3 + 30x^2 + 22x - 11)\sqrt{x^4+4x^3-6x^2+4x+1}\,\Bigr). \end{multline*}

Note that, morally, one expects such integrals to produce inverse elliptic functions and worse, rather than an innocent logarithm of an algebraic function. Abel went on to study, well, abelian integrals, and it is Chebychev who explains more completely --- using continued fractions --- what is going on with these `pseudo-elliptic' integrals. Recently, a student of mine, Xuan Chuong Tran, computed all the $D$ over the rationals of degree $4$ that have an $f$ as above.

I will explain various contexts in which the `certain' polynomials $D$ are of relevance. They include symbolic integration of algebraic functions; the study of units in function fields; diophantine approximation of algebraic numbers; and, given a polynomial $g$, the consideration of period length of the continued fraction expansion of sequences $\sqrt{g(n)}$ as $n$ varies in the integers.

The underlying secret turns out to be the behaviour of the continued fraction expansion of $\sqrt {D(x)}$. So the core of my remarks will be the story of periodic continued fraction expansions both in number fields and in function fields.