### Continued fractions, hyperelliptic curves and higher genus Chebyshev polynomials

Mathematical Physics Seminar

3rd November 2023, 1:45 pm – 3:30 pm

Fry Building, 2.04

We consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g, as originally described by van der Poorten. Using the connection with the classical theory of J-fractions and orthogonal polynomials, we show that in the elliptic case g=1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in particular cases by Chang, Hu, and Xin. We extend these formulae to the higher genus case, and find analogues of Chebyshev polynomials (which correspond to the case g=0). Moreover, for all g>0 we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system, connected with solutions of the infinite Toda lattice.

*Organiser*: Thomas Bothner

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