Independence of continued fractions in function fields

Abstract

A natural abstraction of rationality is that of linear independence, while for algebraicconsideration, a natural abstraction of transcendence is that of algebraic independence. Alsowell-known is that each real number is representable as a simple continued fraction. Combiningindependence with representation, there arises a natural problem of characterizing elements viatheir representations. The work in this thesis centers around these two concepts, namely,continued fraction representation and their independence, in the field of Laurent series over afinite field, referred to here as function field. The major part of the thesis are devoted to the establishing of two general independencecriteria, one for linear and the other for algebraic independence and to the extensivecomputation of intriguing examples. The linear independence criterion states roughly that ifthe partial quotients grow at a moderately fast rate, their continued fractions are linearlyindependent. This linear independence criterion when applied to the real case encompasses thatobtained by Hancl in 2002. As to algebraic independence, a Liouville-type sufficient conditionthrough rational approximations is proved. When applied to continued fractions, it yields anumber of criteria which show that exponentially growing partial quotients imply algebraicindependence. Two interesting types of explicit continued fractions are worked out as examplesillustrating the strength of the criteria so obtained