Hilbert's Sixteenth Problem - The Original Formulation of The Problems

The Original Formulation of The Problems

In his speech, Hilbert presented the problems as:

“ The upper bound of closed and separate branches of an algebraic curve of degree n was decided by Harnack (Mathematische Annalen, 10); 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. (cf. Rohn, Flächen vierter Ordnung, Preissschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig 1886)

 ”

Hilbert continues:

“

Following this purely algebraic problem I would like to raise a question that, it seems to me, can be attacked by the same method of continuous coefficient changing, and whose answer is of similar importance to the topology of the families of curves defined by differential equations - that is the question of the upper bound and position of the Poincaré boundary cycles (cycles limites) for a differential equation of first order on the form:

where X, Y are integer, rational functions of nth degree in resp. x, y, or written homogeneously:

X \left( y {dz \over dt} - z {dy \over dt} \right) + Y\left(z {dx \over dt} - x {dz \over dt} \right) + Z\left(x {dy \over dt} - y {dx \over dt} \right) = 0

where X, Y, Z means integral, rational, homogenic functions of nth degree in x, y, z and the latter are to be considered function of the parameter t.

 ”

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