We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets $U_i$ which are isomorphic to closed smooth hypersurfaces in ${ℂ}^{n+1}$. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety $X\subset {ℂ}^m$ there is a generically-finite (even quasi-finite) polynomial mapping $f:{ℂ}^n\to {ℂ}^m$ such that $X\subset S_f$. This gives (together with [3]) a full characterization of irreducible components of the set $S_f$ for generically-finite polynomial mappings $f:{ℂ}^n\to {ℂ}^m$.

We give a new proof of Miyanishi's theorem on the classification of the additive group scheme actions on the affine plane.

It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber $f^{-1}\left(t₀\right)$ such that r is the Łojasiewicz exponent of grad(f) near the fiber $f^{-1}\left(t₀\right)$. We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula...

For every polynomial F in two complex variables we define the Łojasiewicz exponents ${ℒ}_{p,t}\left(F\right)$ measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents ${ℒ}_{p,t}\left(F\right)$ in terms of the local invariants of singularities of the pencil of projective curves associated with F.

We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds...

We present an example which confirms the assertion of the title.

We study certain problems on polynomial mappings related to the Jacobian conjecture.

Using the notion of uniruledness we indicate a class of algebraic varieties which have a stronger version of the cancellation property. Moreover, we give an affirmative solution to the stable equivalence problem for non-uniruled hypersurfaces.

We study the following cancellation problem over an algebraically closed field of characteristic zero. Let X, Y be affine varieties such that $X{\times }^m\cong Y{\times }^m$ for some m. Assume that X is non-uniruled at infinity. Does it follow that X ≅ Y? We prove a result implying the affirmative answer in case X is either unirational or an algebraic line bundle. However, the general answer is negative and we give as a counterexample some affine surfaces.

We study the behavior at infinity of non-zero constant Jacobian polynomial maps f = (P,Q) in ℂ² by analyzing the influence of the Jacobian condition on the structure of Newton-Puiseux expansions of branches at infinity of level sets of the components. One of the results obtained states that the Jacobian conjecture in ℂ² is true if the Jacobian condition ensures that the restriction of Q to the curve P = 0 has only one pole.

We show that the non-proper value set of a polynomial map (P,Q): ℂ² → ℂ² satisfying the Jacobian condition detD(P,Q) ≡ const ≠ 0, if non-empty, must be a plane curve with one point at infinity.

Using BMY inequality and a Milnor number bound we prove that any algebraic annulus ${ℂ}^{*}$ in ${ℂ}^2$ with no self-intersections can have at most three cuspidal singularities.

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....