Transcendence And Linear Independence Over Function Fields In Positive Characteristic

The transcendency of real or complex numbers is one of the basic problems in number theory.Although almost all real or complex numbers are transcendental,it is in general difficult to know whether a given number is transcendental.Modern number theory tells us that tackling the same problem for rational number field or function fields in positive characteristic,there are many similarities and differences.The present thesis studies linear independence,transcendence,and algebraic independence of formal power series over function fields,and it mainly contains the following four aspects:(1)Linear independence criterion:we present a general linear independence crite-rion for formal power series over function fields in positive characteristic,which contains many existing linear independence criteria and transcendence criteria.As applications,we apply the criterion to show the ?-algebraic independence of ec and 1/?C,the linear independence of special values at rational arguments of the Carlitz exponential function and the Carlitz logarithmic function,and the transcendence of some values of a class of generalized Carlitz-Goss gamma functions.(2)Algebraic independence criterion:we obtain an algebraic independence cri-terion over function fields in positive characteristic for a class of fast convergent power series.By applying this criterion,we show the algebraic independence of Liouville series for function fields in positive characteristic.(3)Transcendence of special values of hypergeometric functions:on one hand,we establish a T-module function equation for a large class of special hypergeometric functions,and obtain some transcendence results about special values of these hypergeo-metric functions,with the help of Schneider-Lang theorem for function fields in positive characteristic.On the other hand,we give a direct proof of weak transcendence of special values of entire hypergeometric functions for function fields in positive characteristic.(4)Four exponentials conjecture:Based on algebraic independence of special values of Carlitz logarithmic function proved by Papanikolas,we show four exponentials conjecture for function fields in positive characteristic,by following the typical approach in the classical transcendental number theory.