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Session 2. Algebraic Geometry
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On the Abhyankar--Moh inequality |
Arkadiusz Płoski, Kielce University of Technology, Poland
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Let \(C\) be a complex affine algebraic curve of degree \(n > 1\)
having only one branch at infinity \(\gamma\) and let \(r_{0},
r_{1},\dots,r_{h}\) be the \(n\)-sequence of the semigroup \(G\) of
the branch \(\gamma\) defined as follows: \(r_{0}=n,
r_{k}=\min\{r\in G: r\not\in\mathbb Nr_{o}+\dots+\mathbb Nr_{k-1}\}
\) for \(k\ge1\) and \(G=\mathbb Nr_{o}+\dots+\mathbb Nr_{h}\).
Then the Abhyankar--Moh inequality (see [1,2]) can be
stated in the form
\begin{equation}\tag{\(AM_{n}\)}\label{equatio}
\gcd\{r_{0},\dots,r_{h-1}\}r_{h}<n^{2}.
\end{equation}
The aim of this talk is to present (see [3]) some results on
the semigrups \(G\subset N\) of plane branches \(\gamma\) with
property \eqref{equatio}. In particular we describe such semigroups
with the maximum conductor.
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References- S.S.Abhyankar, T.T.Moh, Embeddings of the line
in the plane . J. reine angew. Math.276 (1975), 148-166.
- E.García Barroso, A.Płoski, An approach to
plane algebroid branches preprint arXiv:1208.0913 [math.AG].
- R.D.Barrolleta, E.R. García Barroso and A.Płoski,
Appendix to [2] .
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