ANTICIPATION OF HILBERT'S SIXTEENTH PROBLEM

HARNACK, A.

Ueber die Vieltheiligkeit der ebenen algebraischen Curven.

Leipzig, B. G. Teubner, 1876. 8vo. Bound in recent full black cloth with gilt lettering to spine. In "Mathematische Annalen", Volume 10., 1876. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp.189-198. [Entire volume: IV, 592 pp.].

First printing of Harnack's Curve Theorem which formed the background for Hilbert's sixteenth problem.

In real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, describes the possible numbers ofconnected components that an algebraic curve can have, in terms of the degree of the curve.

Hilbert's sixteenth problem was posed as the "Problem of the topology of algebraic curves and surfaces", he presented his problem as follows:
"The upper bound of closed and separate branches of an algebraic curve of degree n was decided by Harnack (the present paper); from this arises the further question as of the relative positions of the branches in the plane.
As of the curves of degree 6, I have - admittedly in a rather elaborate way - convinced myself that the 11 branches, that they can have according to Harnack, never all can be separate, rather there must exist one branch, which have another branch running in its interior and nine branches running in its exterior, or opposite. It seems to me that a thorough investigation of the relative positions of the upper bound for separate branches is of great interest, and similarly the corresponding investigation of the number, shape and position of the sheets of an algebraic surface in space - it is not yet even known, how many sheets a surface of degree 4 in three-dimensional space can maximally have." (Rohn, Flächen vierter Ordnung, Preissschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig 1886).

Hilbert had investigated the M-curves of degree 6, and found that the 11 components always were grouped in a certain way. His challenge to the mathematical community now was to completely investigate the possible configurations of the components of the M-curves.
Furthermore he requested a generalization of Harnack's Theorem to algebraic surfaces and a similar investigation of surfaces with the maximum number of components.
The problem is still unsolved today.

Order-nr.: 47129