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