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Sat, 04 Mar 2000 00:00:00 +0200Sat, 04 Mar 2000 00:00:00 +0200Plane curves of minimal degree with prescribed singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/757
We prove that there exists a positive \(\alpha\) such thatfor any integer \(\mbox{$d\ge 3$}\) and any topological types \(\mbox{$S_1,\dots,S_n$}\) of plane curve singularities, satisfying \(\mbox{$\mu(S_1)+\dots+\mu(S_n)\le\alpha d^2$}\), there exists a reduced irreducible plane curve of degree \(d\) with exactly \(n\) singular points of types \(\mbox{$S_1,\dots,S_n$}\), respectively. This estimate is optimal with respect to theexponent of \(d\). In particular, we prove that for any topological type \(S\) there exists an irreducible polynomial of degree \(\mbox{$d\le 14\sqrt{\mu(S)}$}\) having a singular point of type \(S\).Gert-Martin Greuel; Christoph Lossen; Eugenii Shustinarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/757Mon, 03 Apr 2000 00:00:00 +0200