Systems of polynomial equations are a common occurrence in problems coming from engineering, science, and mathematics. This course aims to provide an introduction to basics of the new area of numerical algebraic geometry that offers effective methods to numerically compute and manipulate solution sets of such systems.

In the first part of the course some background knowledge of algebraic geometry will be briefly reviewed. In particular, we will discuss

• polynomial rings of several variables and their ideals;
• algebraic sets in affine and projective spaces;
• the algebraic solution of a system of polynomials (via computational algebra methods).

The second part will be devoted to the numerical and geometric analysis of solution sets. In particular, we will discuss

• the homotopy continuation method;
• the computation of real or complex solutions of a system of polynomials;
• isolated and positive-dimensional solution sets;
• regular and singular solutions;
• probability-one algorithms (stability and accuracy).

During the course, we will present several examples of applications and we will introduce the dedicated open source software Bertini.