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In this paper we show that the methods of algebraic geometry can be used to study the linear optimal regulator problem. It is shown that under certain conditions almost any system is obtainable by optimal feedback. To do this involves developing general techniques for studying feedback in systems, using methods from the theory of multivariable polynomials. The linear quadratic regulator problem can be viewed as a feedback problem, with feedback preserving the linear symplectic group. New general techniques are developed that might be useful for other systems-theoretic problems; to enhance the possibility of such utilization, a new simpler proof of main "almost-ontoness" theorem from algebraic geometry, using the classical theory of resultants, is given in an Appendix.