The “Sales Tax” Theorem

Abstract

Sometimes in mathematics there are several results which were obtained independently, and at first sight appear unrelated. Then someone finds a more general viewpoint that makes the earlier discoveries special cases of a single new theorem. In the process, it usually becomes clearer what was 'really going in the original results. One such example in the history of mathematics is the underlying similarity between the Chinese remainder theorem and the Lagrange interpolation formula. To state them similarly, the Chinese remainder theorem says: given integers mI, M2,..., mk which are independent in the sense that no two have a common prime factor, and given integer values cI, c2, ... Ck, we can find an integer N which takes the value ci at mi for each i = 1, 2,..., k, in the sense that N ci (mod mi); and N is unique modulo M = mIm2 nMk. Correspondingly, the Lagrange interpolation theorem says: given real numbers x X2, 2... , Xk which are independent (i.e., distinct points) on the real axis, and given real values y1, y2, .. ., yk, we can find a polynomial function f (x) which takes the value yi at xi for each i = 1,2, ..., k, in the sense that f(xi) = yi; and this f(x) is unique for degree less than k. The general result of which these are both special cases is called the Theorem on Independence of Places in algebraic geometry, and requires some specialized knowledge of valuation theory to state properly. In this paper, we shall look at a similar situation, but in this case the general result will be easier to follow than the historical special cases which preceded it. One of these special cases is the following mysterious result in number theory, illustrated in TABLE 1: