Abstract
We describe the structure of the group of algebraic automorphisms of the following surfaces 1) P1 k ×P1 k minus a diagonal; 2) P1 k ×P 1 k minus a fiber. The motivation is to get a new proof of two theorems proven respectively by L. Makar-Limanov and H. Nagao. We also discuss the structure of the semi-group of polynomial proper maps from C2 to C2.
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