Abebaw Tilahun (PhD)Tsegay Hailegebriel2018-07-162023-11-042018-07-162023-11-042011-06http://etd.aau.edu.et/handle/123456789/8764In this Project, we introduced the affine varieties, which are curves and surfaces (and higher-dimensional objects) defined by Polynomial Equations. By considering ideals in the polynomial ring 1,…], we also understood the dimension of an affine variety and the affine Hilbert Function of an ideal, which is a function on the non-negative integers Given an affine variety which is the union of a finite number of linear subspaces of the affine space , we defined the dimension of in terms of the dimensions of the subspaces. We also defined the dimension of in terms of the degree of the affine Hilbert Polynomial of the corresponding ideal . Finally, we stated several basic properties of dimension over an infinite field using the degree of affine Hilbert PolynomialsenDimensions of Affine VarietiesThesis