An Eigenvalue Theorem for Systems of Polynomial Equations

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 finite-dimensional. Everyone is familiar with the representation of a curve in the plane using an algebraic equation such as X + 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 (λ, μ) which are zeros of X+Y −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. Although the theorem is not new (see [2], [4] and [3]) and is quite elementary, we cannot find it in undergraduate text books. If K is any field and f1, . . . , fn are polynomials in the ring K[X,Y ], then it is convenient to consider the ideal J = 〈f1, . . . , fn〉 generated by these polynomials. The set of common zeros in K of f1, . . . , fn is clearly the same as the set of common zeros for all the polynomials in J so we can forget about the particular polynomials chosen to generate J and simply think about the ideal J itself. We shall refer to these zeros briefly as the zeros of J . An important advantage of approaching the problem of the set of common zeros of polynomials (= intersection of curves) in this way is that we can take advantage of the structure of the ring K[X,Y ] rather than simply dealing with a subset of K. At the same time, we should not lose sight of the geometric interpretation of the theorems which arise. In what follows, we shall restrict ourselves to the case of two variables, but at the end of this note we shall point out how all the results can be generalized to the case of m variables X1, . . . , Xm.