We give an overview of recent results concerning kernels of triangular derivations of polynomial rings. In particular, we examine the question of finite generation in dimensions 4, 5, 6, and 7.

We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x+{\left(Ax\right)}^{*3}$ with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with $f\in k^{\left[n\right]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the following:...

Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic and geometric properties of the maps involved. That permits the separate formulation and reduction, though not so far the solution, of the SRJC for classes of nonsingular polynomial endomorphisms of real n-space that exclude the Pinchuk counterexamples to the SRJC,...

Existence of loops for non-injective regular analytic transformations of the real plane is shown. As an application, a criterion for injectivity of a regular analytic transformation of ${ℝ}^2$ in terms of the Jacobian and the first and second order partial derivatives is obtained. This criterion is new even in the special case of polynomial transformations.

Let $F=(f_1,...,f_q)$ be a polynomial dominating map from ${ℂ}^n$ to ${ℂ}^q$. We study the quotient ${𝒯}^1\left(F\right)$ of polynomial 1-forms that are exact along the generic fibres of $F$, by 1-forms of type $\mathrm{d}R+\sum a_i\mathrm{d}{f}_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${𝒯}^1\left(F\right)$ is always a torsion $ℂ[t_1,...,t_q]$-module. Then we determine under which conditions on $F$ we have ${𝒯}^1\left(F\right)=0$. As an application, we study the behaviour of a class of algebraic $({ℂ}^p,+)$-actions on ${ℂ}^n$, and determine in particular when these actions are trivial.

Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B=R^{\left[1\right]}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^{\left[2\right]}$? We give a negative answer to this question. The interest in retracts comes...