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.
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]