Degree Matrices And Computation Of Rational Points

This thesis is composed of four chapters,which mainly study the rational points of a class of hypersurfaces and the number of solutions of the Carlitz equations over finite fields.In the first chapter,we introduce some basic and important results about finite fields and characters,Gauss sums,degree matrices and Smith normal forms,which lay the foundations for the later chapters.In the second chapter,we first introduce a class of hypersurfaces over finite fields,then use the quadratic character and quadratic Gauss sum to obtain the explicit formula for the number of solutions to these hypersurfaces when the greatest invariant factor of degree matrix and -1are not relatively prime where is the order of finite field.This generalizes the results of Sun and Cao.In the third chapter,we first present backgrounds of the Carlitz equations over finite fields,then derive the explicit formula for the number of solutions to these equations when the exponents satisfy certain conditions.This generalizes the results of Baoulina and Cao.In the last chapter,we summarize the paper and put forward some problems.