On the minimal degree of logarithmic vector fields of line arrangements
Résumé
Planar polynomial vector fields may have straight lines as invariant sets and the number of invariant lines they may have, according to the degree of the systems, is a matter of high interest. The quantity can be infinite for any degree, and the cases in which this happens are well known and trivial. So, most interest goes to the cases when the number is finite. If it is not infinite, then the maximum number is bounded by 3n, where n is the degree of the system (counting also the line at infinity). It is also proved that this bound is realizable for any n if we consider simultaneously real and complex invariant straight lines. However, if we limit ourselves to only real straight lines, then for some n's, such as n=2,3,5, the bound is realizable, but for n=4 it is proved that the bound is lower than 3n, and only if we allow also complex lines we can achieve the number 3n. For other n's, the bounds of the number of real invariant straight lines are still unknown.
Simultaneously, one can pose the question: given a concrete arrangement of lines, what is the minimum degree that a vector field must have to contain such a set as invariant? In the paper under review the authors give a negative result concerning a possibility that this degree may be related with the combinatorics of the set of straight lines.