We show that every automorphism of the group ${𝒢}_n:=Aut\left({𝔸}^n\right)$ of polynomial automorphisms of complex affine $n$-space ${𝔸}^n={ℂ}^n$ is inner up to field automorphisms when restricted to the subgroup $T{𝒢}_n$ of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension $n=2$ where all automorphisms are tame: $T{𝒢}_2={𝒢}_2$. The methods are different, based on arguments from algebraic group actions.

We charocterize the commuting polynomial automorphisms of C2, using their meromorphic extension to P2 and looking at their dynamics on the line at infinity.

Let X be a smooth algebraic hypersurface in ℂⁿ. There is a proper polynomial mapping F: ℂⁿ → ℂⁿ, such that the set of ramification values of F contains the hypersurface X.

We generalize some results on reconstructing sets to the case of ideals of 𝕜[X₁,...,Xₙ]. We show that reconstructing sets can be approximated by finite subsets having the property of reconstructing automorphisms of bounded degree.

The equivalence of the definitions of the Łojasiewicz exponent introduced by Ha and by Chądzyński and Krasiński is proved. Moreover we show that if the above exponents are less than -1 then they are attained at a curve meromorphic at infinity.

Let $h=\sum h_{\alpha \beta }{X}^{\alpha }{Y}^{\beta }$ be a polynomial with complex coefficients. The Łojasiewicz exponent of the gradient of h at infinity is the least upper bound of the set of all real λ such that ${\left|gradh(x,y)\right|\ge c\left|(x,y)\right|}^{\lambda }$ in a neighbourhood of infinity in ℂ², for c > 0. We estimate this quantity in terms of the Newton diagram of h. Equality is obtained in the nondegenerate case.

L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T. Yu solved the stable equivalence problem for the polynomial ring k[x,y] when k is a field of characteristic 0. In this note we give an affirmative solution for an arbitrary field k.