Final polynomials allow us to view the question of non-realizablility of oriented matroids in the algebraic setting of bracket algebras. A final polynomial exists for every non-realizable oriented matroid and can provide a concise proof for non-realizability.
In this talk I will define final polynomials and bi-quadratic final polynomials. This is the first of two talks based on "Euclideanness and final polynomials in matroid theory" by Jürgen Richter-Gebert.
In the second talk, I will show how to find biquadratic final polynomials for non-euclidean oriented matroids.