Periodic Points on an Algebraic Variety

where the Li(x) are homogeneous polynomials of degree 1 with coefficients which are algebraic numbers. One of the most interesting cases arises when the following two conditions are satisfied. (1) The Li(x) never vanish simultaneously at any point of V. (2) Under the mapping a point of V again becomes a point of V. Generally speaking the only points which interest us are those which can be expressed with coordinates which are algebraic numbers. If P = (to, I, * * * n is such a point, the number field generated by the ratios ti/tj will be the smallest field in which P is rational. The degree of this field over the rational numbers, we shall call the degree of rationality of P. Suppose now that conditions (1) and (2) are satisfied, then by iterating the mapping, a given point P of V will generate an infinite sequence P = Po, P1, P2, *** of points on the variety. This sequence will consist either of distinct points or else there will be repetitions. In the latter case the sequence ultimately becomes periodic, and we shall describe this situation by saying "P is an exceptional point". One of the results established can now be stated, namely, "If we exclude the case in which the mapping is linear, then there will be at most a finite number of exceptional points with a given degree of rationality". We can illustrate this result by means of the well known Weierstrass function P(Z1 g2, g3), where g2 and g3 are algebraic numbers. A complex number 'a' is called a "division point" if for some integer n, which is not zero, na is a period. The values of p(z) at the division points are called "division values", and it is easy to see that these are algebraic numbers. By comparing the elliptic function with the exponential function, we see that these division values are in some respects analogous to roots of unity, and it is therefore natural to inquire how they are distributed among the number fields. A special case of the general theorem stated above, shows that if we limit ourselves to number fields whose absolute degrees are bounded by some given integer, then amongst all these fields there will be only a finite number of division values. These facts are consequences of Theorem 2 which is an inequality proved under very general conditions. This inequality has another application (not discussed here), namely 167