General non-realizability certificates for spheres with linear programming

In this paper we present a simple technique to derive certificates of non-realizability for combinatorial polytope. Our approach uses variant the classical algebraic introduced by Bokowski and Sturmfels (1989), final polynomials. More specifically reduce problem finding realization that positive point in variety try find polynomial with coefficients generating ideal (a polynomial), showing such does not exist. Many, if most, techniques proving developed last three decades can be seen as following framework, using more or less elaborate ways constructing proposal is straightforward simply use linear programming exhaustively search polynomials restricted some subspace. Somewhat surprisingly, elementary strategy yields results are competitive alternatives, allows us new examples non-realizable polytopes.