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Using the theory of minimal models of quasi-projective surfaces we give a newproof of the theorem of Lin-Zaidenberg which says that every topologicallycontractible algebraic curve in the complex affine plane has equation $X^n=Y^m$in some algebraic coordinates on the plane. This gives also a proof of thetheorem of Abhyankar-Moh-Suzuki concerning embeddings of the complex line intothe plane. Independently, we show how to deduce the latter theorem from basicproperties of $\mathbb{Q}$-acyclic surfaces.

A new proof of the theorems of Lin-Zaidenberg and Abhyankar-Moh-Suzuki