In this article, we introduce a special class of non complete webs, the NN-webs. We also study the algebraic and geometric properties of these webs.

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 G be a complex affine algebraic group and H,F ⊂ G be closed subgroups. The homogeneous space G/H can be equipped with the structure of a smooth quasiprojective variety. The situation is different for double coset varieties F∖∖G//H. We give examples showing that the variety F∖∖G//H does not necessarily exist. We also address the question of existence of F∖∖G//H in the category of constructible spaces and show that under sufficiently general assumptions F∖∖G//H does exist as a constructible space....

We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity $\left(J_{\infty }\right)$: There does not exist a sequence $(p_k,q_k)\subset ℂ²$ of complex singular points of F such that the imaginary parts $(\Im \left(p_k\right),\Im \left(q_k\right))$ tend to (0,0), the real parts $(\Re \left(p_k\right),\Re \left(q_k\right))$ tend to ∞ and $F(\Re \left(p_k\right),\Re \left(q_k\right)))\to a\in ℝ²$. It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition $\left(J_{\infty }\right)$ and if, in addition, the restriction of F to every real level set $P^{-1}\left(c\right)$ is proper for values of |c| large enough.

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\overline{X}\to X$$ of X as a quotient of a vector space $$\overline{X}$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

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.

We show that the GVC (generalized vanishing conjecture) holds for the differential operator $\Lambda =({\partial }_x-\Phi \left({\partial }_y\right)){\partial }_y$ and all polynomials $P(x,y)$, where $\Phi \left(t\right)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

We associate to a given polynomial map from ${ℂ}^2$ to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

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.

We give the formula expressing the Łojasiewicz exponent near the fibre of polynomial mappings in two variables in terms of the Puiseux expansions at infinity of the fibre.