Nonrealizability proofs in computational geometry (Q912857)

This paper is devoted to the study of some problems concerning real algebraic geometry. By using the concept of final polynomials as an approach to prove nonrealizability for oriented matroids and combinatorial geometries, the authors describe an algorithm for constructing such final polynomials for a large class of nonrealizable chirotopes. An example to show that not every realizable simplicial chirotope admits a solvability sequence is performed in the paper. This means that it is not so easy to use a combinatorial method for proving nonrealizability, therefore the algorithm given in the paper is very interesting. For oriented matroids, some main results in the paper are the following ones: 1. Let M be a matroid and K be a field. Then one and only one of the following statements is true: a. There exists a final polynomial for M with coefficients in K. b. M is realizable over some finite algebraic field extension of K. 2. A matroid M is not realizable over an algebraically closed field K if and only if there exists a final polynomial for M over K.