Computational Tutorial on Gröbner bases embedding Sage in LaTeX with SageTEX

A comprehensive treatment of the theory of Gröbner bases is far beyond the scope of this tutorial. The reader is reffered to [1], [2], [3] for a more detailed discussion. Gröbner bases offers a powerful algorithmic criterion for the existence of solutions to a system of algebraic equations over algebraically closed fields. While the theory is rich and builds on the theory of Ideals and Varieties, the discussion in the persent tutorial will remain as elementary as possible. The tutorial focuses on implementation aspects of Gröbner bases computation via the Buchberger algorithm. The Sage[S11] specific implementation discussed here is not meant to rival the efficient implementations available on Singular, Maxima and other Computer Algebra Systems(CAS). The tutorial merely attempts to provide an overview of the implementation details of the Buchberger algorithm. It must be pointed out that there are many other algorithms besides the Buchberger algorithm for computing Gröbner bases, however for simplicity we restrict our attention to the Buchberger Algorithm. The reader can contact the author 1 , to obtain the LTEX source or to suggest possible corrections The problem that the tutorial discusses is the determination of existence of solutions to a system of polynomial equations of the form.