Systems of Functions

In the theory of algebraic invariants, questions as to the finiteness of complete systems of forms deserve, as it seems to me, particular interest. L. Maurer has lately succeeded in extending the theorems on finiteness in invariant theory proved by P. Gordan, to the case where, instead of the general projective group, any subgroup is chosen as the basis for the definition of invariants.

An important step in this direction had been taken al ready by A. Hurwitz, who, by an ingenious process, succeeded in effecting the proof, in its entire generality, of the finiteness of the system of orthogonal invariants of an arbitrary ground form.

The study of the question as to the finiteness of invariants has led me to a simple problem which includes that question as a particular case and whose solution probably requires a decidedly more minutely detailed study of the theory of elimination and of Kronecker’s algebraic modular systems than has yet been made.

Let a number m of integral rational functions Xl, X2, … , Xm, of the n variables xl, x2, … , xn be given,

(S)   X1 = f1(x1, … , xn),
X2 = f2(x1, … , xn),

Xm = fm(x1, … , xn).

Every rational integral combination of Xl, … , Xm must evidently always become, after substitution of the above expressions, a rational integral function of xl, … , xn. Nevertheless, there may well be rational fractional functions of Xl, … , Xm which, by the operation of the substitution S, become integral functions in xl, … , xn. Every such rational function of Xl, … , Xm, which becomes integral in xl, … , xn after the application of the substitution S, I propose to call a relatively integral function of Xl, … , Xm. Every integral function of Xl, … , Xm is evidently also relatively integral; further the sum, difference and product of relative integral functions are themselves relatively integral.

The resulting problem is now to decide whether it is always possible to find a finite system of relatively integral function Xl, … , Xm by which every other relatively integral function of Xl, … , Xm may be expressed rationally and integrally.

We can formulate the problem still more simply if we introduce the idea of a finite field of integrality. By a finite field of integrality I mean a system of functions from which a finite number of functions can be chosen, in terms of which all other functions of the system are rationally and integrally expressible. Our problem amounts, then, to this: to show that all relatively integral functions of any given domain of rationality always constitute a finite field of integrality.

It naturally occurs to us also to refine the problem by restrictions drawn from number theory, by assuming the coefficients of the given functions fl, … , fm to be integers and including among the relatively integral functions of Xl, … , Xm only such rational functions of these arguments as become, by the application of the substitutions S, rational integral functions of xl, … , xn with rational integral coefficients.

The following is a simple particular case of this refined problem: Let m integral rational functions Xl, … , Xm of one variable x with integral rational coefficients, and a prime number p be given. Consider the system of those integral rational functions of x which can be expressed in the form

G(Xl, … , Xm) / ph,

where G is a rational integral function of the arguments Xl, … , Xm and ph is any power of the prime number p. Earlier investigations of mine show immediately that all such expressions for a fixed exponent h form a finite domain of integrality. But the question here is whether the same is true for all exponents h, i. e., whether a finite number of such expressions can be chosen by means of which for every exponent h every other expression of that form is integrally and rationally expressible.