Diophantine Approximation in Finite Characteristic [chapter]

Dinesh S. Thakur
2004 Algebra, Arithmetic and Geometry with Applications  
In contrast to Roth's theorem that all algebraic irrational real numbers have approximation exponent two, the distribution of the exponents for the function field counterparts is not even conjecturally understood. We describe some recent progress made on this issue. An explicit continued fraction is not known even for a single non-quadratic algebraic real number. We provide many families of explicit continued fractions, equations and exponents for non-quadratic algebraic laurent series in
more » ... ent series in finite characteristic, including non-Riccati examples with both bounded or unbounded sequences of partial quotients. On this occasion of Professor Abhyankar's 70th birthday conference, it might be appropriate to mention some recent applications of the 'high school algebra' [A] to the study of diophantine approximation for function fields in finite characteristic. This study is related to some of his loves: power series, continued fractions, algebraic curves, finite characteristic, resultants (and even automata).
doi:10.1007/978-3-642-18487-1_46 fatcat:k2dmefsxvvbitdq26jeqfozfvm