This paper is concerned with problems of generic stabilizability of time- invariant linear systems by static linear output feedback. Let \(L_{m,n,p}\) be the space of all real linear systems (A,B,C) having m inputs, n states and p outputs. The problem is: For which dimensions m,n,p can one stabilize all (A,B,C) \(\in L_{m,n,p}\) except perhaps those contained in a proper algebraic set by some output feedback \(K\in {\mathbb{R}}^{mxp}?\) Using algebraic geometric methods the authors show that mp\(\geq n\) is necessary for generic stabilizability. The proof is based on another interesting result of this paper which says that, for \(mp=n\), generic stabilizability is equivalent to the condition that for all real \(\rho\) and the generic (A,B,C) \(\in L_{m,n,p}\) there exists a gain \(K\in {\mathbb{R}}^{mxp}\) such that the closed loop characteristic polynomial is \((s\)-p)\({}^ n\). Finally, methods of decision algebra are applied to study the question under which conditions rational algorithms exist for finding a gain K which places the poles or stabilizes the system.