Geometry, Algebra, and Algorithms

Abstract

This chapter will introduce some of the basic themes of the book. The geometry we are interested in concerns affine varieties, which are curves and surfaces (and higher dimensional objects) defined by polynomial equations. To understand affine varieties, we will need some algebra, and in particular, we will need to study ideals in the polynomial ring k[x1,{\thinspace}…,{\thinspace}xn]. Finally, we will discuss polynomials in one variable to illustrate the role played by algorithms.