Definable sets over finite fields

Counting the number of points of an algebraic set over a finite field has been studied by Hasse, Lang and Weil, and is an important theme in algebra. In this thesis, we present the results found by Chatzidakis, van den Dries and Macintyre in the article "Definable Sets over Finite Fields" and their applications. These results give estimates of the number of points of definable sets over finite fields. Main theorem of the thesis says that given a formula with $n$ variables, the number of points of the set defined by this formula in a finite field $\mathbb{F}_q$ with $q$ elements is approximately $\mu q^d$. The constants $\mu$ and $d$ can take only finitely many values independent of the field $\mathbb{F}_q$ the formula is defined in.