Book excerpt: This work deals with the construction of zero mean curvature surfaces in the three dimensional anti de-Sitter space. The method that was used was to look for examples that can be described using polynomial equations. More precisely, we are looking for "algebraic" examples of zero mean curvature surfaces. Success was found in finding new examples. Actually the author proved that if we used some polynomial equations previously discovered by Blaine Lawson to study a similar problem in the three dimensional sphere, these equations will also produce examples with zero mean curvature in the three dimensional anti de-Sitter space. This is considered the main result of the thesis. The family of examples that were found is so rich that it shows the existence of algebraic examples of any order; that is, examples produced with equations involving a polynomial of any desire degree. This thesis is written to be as self-contained as possible. In regard with the method used to prove the main result, two different approaches are explained to prove that a particular surface has zero mean curvature. One of them uses a parametrization of the surface and the other uses the polynomial that describes the surface. Even though both methods were thoroughly studied in the thesis, the former one was used to prove the main result.