An Eigenvalue Theorem for Systems of

We give a new short proof of a theorem relating solutions of a system of polynomial equations to the eigenvalues of the multiplication operators on the quotient ring, in the case when the quotient ring is …nite-dimensional. Everyone is familiar with the representation of a curve in the plane using an algebraic equation such as X 2 +Y 2 = 1. Algebraic geometers have learned that it is more convenient to represent these curves in the equivalent form of solutions (or zeros) of polynomials. In the case above, the circle is simply the set of (�;�) which are zeros of X 2 +Y 2 1. Similarly the simultaneous zeros of several polynomials represent the points of intersection of all of the corresponding curves. The solution of a system of polynomial equations is a natural extension of the problem of solving linear equations, and arises, for example, in the Lagrange multiplier method when the constraints and the function to be optimized are algebraic. Our particular aim is to prove the theorem below (Theorem 4) which gives a description of the common zeros of a system of polynomials. It is interesting in itself because it combines various important concepts and results from a standard undergraduate curriculum: ideals, quotient rings, homomorphism theorems, commuting linear operators and their common eigenvectors.