Diophantine Approximation And Continued Fractions In Power Series Fields

Defining an ultrametric absolute value in the formal power series K((T-1)),which then can be considered as the completed field of the function field K(T),just as R is the completed of Q for the usual absolute value.We just suppose that K is a finite field in most parts of this report and we denote the formal power series by F(q)if K is a finite field with q elements,where q is a power of prime p.In Chapter 2 we can see many questions in number theory which have been studied in the setting of the real numbers can be transposed the setting of the power series.We also present the argument of Liouville producing transcendental real numbers and its formal case version,Mahler's theorem.Hyperquadratic elements is a particular subset of F(q),and we denote this set by H(q).These power series are irrational elements ??F(q)satisfying an equation? = f(?T)where r is a power of the characteristic p of the base field and f is a linear fractional transformation with integer coefficients(polynomial in Fq[T]).When Mahler gave his formal version,he also gave an example which is hyperquadratic and the example can indicate that Roth's theorem in real case does not work in the formal case.We will study the properties of hyperquadratic elements in Chapter 2 and Chapter 3.In mathematics,a continued fraction is an expression obtained through an itera-tive process of representing a number as the sum of its integer part and the reciprocal of another number,then writing this other number as the sum of its integer part and another reciprocal,and so on.At first,continued fractions were considered for real numbers.In the 1930's,L.Carlitz initiated formal arithmetic,replacing the real num-bers by polynomials over a finite field.The irrationality measure of a real number or a formal number is a measure of how closely it can be approximated by rationals.We discuss the continued fractions process and irrationality in both real and formal cases in Chapter 3.Continued fractions are the tool to study rational approximation of numbers of functions,it offer a means of concrete representation for arbitrary real numbers and formal power series over finite fields.Given a particular irrational element,the continued fraction expansion is seldom known by people.In the 1930's,Kintchine put forward a question concerning these algebraic elements in R of degree>3 over Q:is the sequence of partial quotients unbounded?It is a still open problem.But for formal power series the situation is different.In Chapter 4 we introduce the Voloch's theorem which states that if ? ?K((T-1))is hyperquadratic,where K is a field of positive characteristic,then a has bounded partial quotients if and only if v(?)= 2,where u(?)denotes the irrationality measure of ?.Note that elements if ? ? F(q)of degree 2 or 3,then easily we get ??H(q).In Chapter 5 we introduce the work of Bluher and Lasjaunias to obtain some necessary conditions for quartic elements to be a hyperquadratic element.My main aim in this report has been to study the basic theory.There are no new results here;all are known results.I would like to thank Professor Alain Lasjaunias,for numerous helpful discus-sions and I want to thank Professor Qing Liu and the program "MathBoX" for bring-ing me to this stage.