The following is a simple particular case of this refined problem: Let integral rational functions of one variable with integral rational coefficients, and a prime number be given. Consider the system of those integral[Pg 28] rational functions of which can be expressed in the form where is a rational integral function of the arguments and is any power of the prime number . Earlier investigations of mine[34] show immediately that all such expressions for a fixed exponent form a finite domain of integrality. But the question here is whether the same is true for all exponents , i. e., whether a finite number of such expressions can be chosen by means of which for every exponent every other expression of that form is integrally and rationally expressible. From the boundary region between algebra and geometry, I will mention two problems. The one concerns enumerative geometry and the other the topology of algebraic curves and surfaces. [32] Cf. Sitzungsber. d. K. Acad. d. Wiss. zu München, 1890, and an article about to appear in the Math. Annalen. [33] "Ueber die Erzeugung der Invarianten durch Integration," Nachrichten d. K. Geseltschaft d. Wiss. zu Göttingen, 1897. [34] Math. Annalen, vol. 36 (1890), p. 485. 15. RIGOROUS FOUNDATION OF SCHUBERT'S ENUMERATIVE CALCULUS. The problem consists in this: To establish rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert[35] especially has determined on the basis of the so-called principle of special position, or conservation of number, by means of the enumerative calculus developed by him. Although the algebra of to-day guarantees, in principle, the possibility of carrying out the processes of elimination, yet for the proof of the theorems of enumerative geometry decidedly more is requisite, namely, the actual carrying out of the process of elimination in the case of equations of special form in such a way that the degree of the final equations and the multiplicity of their solutions may be foreseen. [35] Kalkül der abzählenden Geometrie, Leipzig, 1879. 16. PROBLEM OF THE TOPOLOGY OF ALGEBRAIC CURVES AND SURFACES.