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What are the automorphisms of the 2-variable polynomial ring over $\mathbb{F}_2$, the field with 2 elements? Are they generated by $(x \mapsto y, y \mapsto x)$, $(x\mapsto x+ p(y), y\mapsto y)$, and $(x \mapsto x, y \mapsto y + p(x))$ where $p$ runs over all polynomials over $\mathbb{F}_2$? These are automorphisms, right?

I can see that any automorphism must fix the constants, but not much more.

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    The keywor$d$ here is "affine Cremona group." The answer might be known for two variables but I think not much is known in general.2012-09-18

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(I updated my comment to an answer, so that it doesn't get lost)

You are right, the group of automorphisms is generated by the two types of automorphisms you suggested. It follows from Jung - van der Kulk theorem (for a general field you need affine transformations as generators too, but for $\mathbb{F}_2$ they are already generated by your automorphisms). For a precise statement valid over any field see sect 2.3 and 2.4 of http://www.ams.org/journals/tran/1992-331-01/S0002-9947-1992-1038019-2/S0002-9947-1992-1038019-2.pdf .

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    Nice. Should have thought o$f$ that.2012-10-02