Mathematical Problems

The maximum number of closed and separate branches which a plane algebraic curve of the th order can have has been determined by Harnack.[36] There arises the further question as to the relative position of the branches in the plane. As to curves of the th order, I have satisfied myself—by a complicated process, it is true—that of the eleven blanches which they can have according to Harnack, by no means all can lie external to one another, but that one branch must exist in whose interior one branch and in whose exterior nine branches lie, or inversely. A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space. Till now, indeed, it is not even known what is the maximum number of sheets which a surface of the th order in three dimensional space can really have.[37]

In connection with this purely algebraic problem, I wish to bring forward a question which, it seems to me, may be attacked by the same method of continuous variation of coefficients, and whose answer is of corresponding value for the topology of families of curves defined by differential equations. This is the question as to the maximum number and position of Poincaré's boundary cycles (cycles limites) for a differential equation of the first order and degree of the form where and are rational integral functions of the th degree in and . Written homogeneously, this is where , and are rational integral homogeneous functions of the th degree in , and the latter are to be determined as functions of the parameter .