Euclideaness and final polynomials in oriented matroid theory

This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial 3], 5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel 6], 7] concerning non-degenerate cycling of linear programs in non-euclidean oriented matroids.