内容简介
本书是Springer“数学讲义丛书”之536卷，作者是美国科罗拉多州立大学数学系教授，本书基于M.Ratliff和K.Spackman在该校授课的讲义编写而成。本书由浅入深，以很简单的曲线方程yd=f(x)开篇，介绍了有限域上的各种方程。读者对象：数学专业的研究生和科研工作者.
目录
IntroductionI.Equations yd=f(x)and yq-y=f(x) 1.Finite Fields 2.Equations yd=t(x) 3.Construction of certain polynomiale 4.Proof of the Main Theorem 5.Removal of the condition (m,d)=1 6.Hyperderivatives 7.Removal of the condition that q=p or p2 . 9.Equations yq-y=f(x)II.Character Sums and Exponential Sums 1.Characters of Finite Abelian Groups 2.Characters and Character Sums associated with Finite Flelds 3.Gaussian sums 4.The low road. 5.Systems of equations y1d1=f1(x), ,yndn=fn（x）. 6.Auxiliary lemmas on wv1+…+□ 7.Further auxillary lemmas 8.Zeta Function and L-Functions 9.special L-Functions 10.Fleld extengiong.The Davenport-Hasse relations 11.Proof of the Principal Theorems 12.Kloosterman Sums 13.Further ResultsIII.Abeolutoly Irreducible Equatione f(x,y)=0 1.Introduction 2.Independence results 3.Derivatives. 4.Construction of two algebraic functions 5.Constructlon of two polynomlals 6.Proof of the Main Theorem 8.Hyperderivatives again 9.Removal of the condition that q=pIV.Equations in Many Variables 1.Theorems of Chevalley and Warning 2.Quadratic forms 3.Elementary upper bounds.Projective zeros 4.The average number of zeros of a polynomial 5.itive Bquatione:A Chebychev Argument 6.itive Equations:Character Sums 7.Bquations f,(y）x,l1+ +f(y）xn=oV.Absolutely Irreducible squations f(x,.…,x)=0 1.Elimination Theory 2.The absolute irreducibility of polynomlals(I) 3.The abaolute irreducibllity of polynomial8(II) 4.The absolute 1rreducibility of polynomlals(III) 5.The number of zeroe of abaolutely irreducib1e polynomiale in n varlablesVI.Rudimente of Algebraic Goometry,The Number of Pointa in Varieties over Finite Fields 1.Varietoes 2.Dimension 3.Rational Maps 4.BirationalMaps 5.Linear Disjointness of Fields 6.Constant Field Extensions 7.Counting Points in Varieties Over Finite Pields
