Solubility Of Some Equations In Finite Fields

The so-called indefinite equation,it is to show the number of unknown more than equations of equation(or equations), The theory of indefinite equation, usaully limit unknown to some given sets.In this thesis,we mainly study the existence and the number of soulation of some types equations.Full text content is divided into three parts.In the first chapter,we introduce the background and significance of the research of the problem,and the main results by the author.In the second chapter,we first review the definition and properties of character sum, and introduce some results related to our subj ect,by the known character sum estimates,we have reached the following two results:定理2.6let p be a prime,and A1,A2(?)Zp,│A1││A2│≥10p. Then there are x,y,z∈Zp,such that x+y+z∈A1,xyz∈A2.定理2.7let p be a prime,A1,A2(?)Zp,A(?)Zp*be a multiplicative subgroup,│A│2│A1│p│A2│≥13p3.Then there are x,y,z∈Zp,such that x∈A,x+y+z∈A1,xyz∈A2.These results generalize the Shkredov’s results(Math,Note,88(2010),4:603-611).In the third chapter,We mainly using F.urier transform,then get the number of soulation of equations x1k+…+xsk=(x1+…+xs)2, xi∈Ai,i=1,…,s andy1k+y2k+x3k+…+xs.=mk, y1∈C1,y2∈C2,xi∈Ai,i=3,…s,m∈Fq*over finite field.