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.

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.

We obtain, in a simple way, an estimate for the Noether exponent of an ideal I without embedded components (i.e. we estimate the smallest number μ such that ${\left(radI\right)}^{\mu }\subset I$).

Given a real n×n matrix A, we make some conjectures and prove partial results about the range of the function that maps the n-tuple x into the entrywise kth power of the n-tuple Ax. This is of interest in the study of the Jacobian Conjecture.

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.