ICTP/SISSA Joint Colloquium. Straight lines of a plane and real algebraic geometry
Starts 23 Jan 2008 15:00
Ends 23 Jan 2008 20:00
Central European Time
ICTP
Main Building Main Lecture Hall
Strada Costiera, 11
I - 34151 Trieste (Italy)
Which values may attain the number $M$ of the connected components of the complement to $n$ distinct straight lines in the real plane?
For $n=4$ the possible numbers of components are (5, 8, 9, 10, 11), but the general problem for $n$ straight lines is unsolved.
Namely, between the minimal number of components, $n+1$, and
the maximal one, $(n^2+n+2)/2$, there are gaps formed by the
unattainable numbers.
The first gap is $n+1<M<2n$: these values are unattainable.
For the $k$-th gap the stable boundaries $a_k(n)<M< b_k(n)$ are known, providing the answer for the stable case (where the number of straight lines is sufficiently high, $n>C(k)$). For the unstable values of $n$, the $k$-th gap may be smaller than the stable answer. Its exact boundary is unknown even for the 3-rd gap (where the stability starts from $n=14$ straight lines).
In real algebraic geometry even the simplest problems are difficult (and even the contribution of Hilbert to his 16-th problem, discussing these questions, was wrong).