Systems of polynomial equations occur frequently in applications, especially engineering applications. It is therefore important to have good methods to solve these equations. Three important requirements for a method to be considered useful are, first, that it finds all the solutions: second, that it finds them all accurately; and, finally, that it finds them all quickly. It has recently been observed that, contrary to how students are first taught in linear algebra, it is useful to convert a polynomial problem to an eigenvalue problem. This is true in part because polynomials are sensitive to small changes in their input data, whereas matrices are usually much less sensitive, but conversion to eigenvalue problems is also useful in the multivariate case because good numerical methods are available to solve numerically the generalized eigenproblems that result.

In this talk, I will give a short introduction to the Bezout Matrix and the companion matrix pencil. Finally, I will introduce an efficient method to solve a (non-linear) bivariate polynomial system, based on interpretation of resultants as eigenvalue problems. We show some examples at the end.

This work is under the supervision of Professor Robert M. Corless (Applied Math, University of Western Ontario). Professor Laureano Gonzalez-Vega and Dr. Gema Diaz-Toca of the University of Cantabria, Spain, have been very helpful.