Given an unstable finite-dimensional linear system, one can relate the existence of a memoryless feedback law stabilizing the system to the existence of a real solution of a set of multivariable polynomial inequalities. From these inequalities, a set of equalities may be constructed with two properties: the equality set has a real solution precisely when the inequality set does; generically the equality set has a finite number of solutions. Multivariable polynomial resultants provide a method of solving the equalities subject to the condition that the equalities have a finite number of solutions. The property that there is a finite number of solutions is established using some results of algebraic geometry.
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Anderson et al. (1977) studied this question.