Using Groebner Basis Computations to Teach Algebra

Michael Bulmer
mrb@maths.uq.edu.au
Mathematics
University of Queensland
University of Queensland
Australia

Abstract

Students usually leave secondary school with skills in solving simultaneous equations, even systems with polynomial equations such as is needed for finding the intersections of a line with a circle. Examples like this give an understanding that a system of equations may have no solution, or may have more than one solution. However, it is rare that students will work with polynomial systems which are underdetermined, where they can not actually “solve” the equations but must instead find some reduced form. In early tertiary life students then learn about matrix algebra and vector spaces, seeing a general approach to solving systems of linear equations, including the subspaces generated by underdetermined systems. But little is then done about systems with higher degree polynomials, except perhaps the numerical solution of specific systems. This is partly due to the fact that a new language is required, replacing vector spaces with ideals and varieties. However, there is a simple way to introduce these ideas. Computer algebra systems like Mathematica and Maple include functions for solving systems of polynomial equations. An excellent means of understanding something is to try and teach it to someone else, or something else, in the case of a computer. So how would we teach a computer to solve polynomial equations? This is an ideal reflective exercise for students to think about the algebraic skills they have come to take for granted. They quickly see that there are two steps involved: the simplification of the system and then the application of rules like the quadratic formula or numerical methods when you have the simpler form. For underdetermined systems, we can simply leave out the second step in this process and concentrate on the simplification. Again the need to teach a computer how to do this motivates a deeper understanding of the algebra involved, and leads to a description of Groebner bases and Buchberger’s algorithm. A natural application of these ideas, again drawing on a student’s background, is in the automated proof of geometry theorems. This is naturally an underdetermined setting since we want to prove general theorems. This approach has been used with tertiary students having a range of backgrounds. Experiences and student feedback will be presented.


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