Let k be a field of characteristic zero and $k^{[n]}$ the polynomial algebra in n variables over k. The LND conjecture concerning the images of locally nilpotent derivations arose from the Jacobian conjecture. We give a positive answer to the LND conjecture in several cases. More precisely, we prove that the images of rank-one locally nilpotent derivations of $k^{[n]}$ acting on principal ideals are MZ-subspaces for any $n\geq 2$, and that the images of a large class of locally nilpotent derivations of $k^{[3]}$ (including all rank-two and homogeneous rank-three locally nilpotent derivations) acting on principal ideals are MZ-subspaces.

We study the possible dynamical degrees of automorphisms of the affine space $\mathbb {A}^n$ . In dimension $n=3$ , we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb {A}^n$ for some n, and we give the best possible n for quadratic integers, which is either $3$ or $4$ .

A self-map F of an affine space ${\bf A}_k^n $ over a field k is said to be a Keller map if F is given by polynomials F1, …, Fn ∈ k[X1, …, Xn] whose Jacobian determinant lies in $k\setminus \{0\}$. We consider char(k) = 0 and assume, as we may, that the Fis vanish at the origin. In this note, we prove that if F is Keller then its base ideal IF = 〈F1, …, Fn〉 is radical (a finite intersection of maximal ideals in this case). We then conjecture that IF = 〈X1, …, Xn〉, which we show to be equivalent to the classical Jacobian Conjecture. In addition, among other remarks, we observe that every complex Keller map admits a well-defined multidimensional global residue function.

The factorial conjecture was proposed by van den Essen et al. [‘On the image conjecture’, J. Algebra 340(1) (2011), 211–224] to study the image conjecture, which arose from the Jacobian conjecture. We show that the factorial conjecture holds for all homogeneous polynomials in two variables. We also give a variation of the result and use it to show that the image of any linear locally nilpotent derivation of $\mathbb{C}[x,y,z]$ is a Mathieu–Zhao subspace.

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.

The algebra of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra is isomorphic to the algebra

of scalar integro-differential operators provided that char(K) = 0. Ignoring the non-Noetherian property, the algebra belongs to a family of algebras like the nth Weyl algebra An and the polynomial algebra P2n. Explicit generators are found for the group Gn of automorphisms of the algebra and for the group of units of (both groups are huge). An analogue of the Jacobian homomorphism AutK-alg (Pn) → K* is introduced for the group Gn (notice that the algebra is non-commutative and neither left nor right Noetherian). The polynomial Jacobian homomorphism is unique. Its analogue is also unique for n > 2 but for n = 1, 2 there are exactly two of them. The proof is based on the following theorem that is proved in the paper:

In this note we identify the classes of $\text{Q}$-homological planes in the Grothendieck group of complex varieties ${{K}_{0}}\left( \text{Va}{{\text{r}}_{\text{C}}} \right)$. Precisely, we prove that a connected, smooth, affine, complex, algebraic surface $X$ is a $\text{Q}$-homological plane if and only if $\left[ X \right]\,=\,\left[ \text{A}_{\text{C}}^{2} \right]$ in the ring ${{K}_{0}}\left( \text{Va}{{\text{r}}_{\text{C}}} \right)$ and $\text{Pic}{{\left( X \right)}_{\text{Q}}}\,:=\,\text{Pic}\left( X \right)\,{{\otimes }_{\text{Z}}}\,\text{Q}\,\text{=}\,\text{0}$.

In this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. Let $A$ be a finitely generated commutative $K$–algebra over a field of characteristic 0, and let $\sigma$ be a $K$–algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that ${{\sigma }^{m}}(I)\,\supseteq \,J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \,\in \,\text{Au}{{\text{t}}_{k}}(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with ${{\sigma }^{m}}(Z)\,\subseteq \,Y$ is as above. We present examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic.

Explicit generators are found for the group G2 of automorphisms of the algebra of one-sided inverses of a polynomial algebra in two variables over a field. Moreover, it is proved that

where S2 is the symmetric group, is the 2-dimensional algebraic torus, E∞() is the subgroup of GL∞() generated by the elementary matrices. In the proof, we use and prove several results on the index of an operator. The final argument is the proof of the fact that K1() ≃ K*. The algebras and are noncommutative, non-Noetherian, and not domains.

We strengthen certain results concerning actions of $\left( \mathbb{C},\,+ \right)$ on ${{\mathbb{C}}^{3}}$ and embeddings of ${{\mathbb{C}}^{2}}$ in ${{\mathbb{C}}^{3}}$, and show that these results are in fact valid over any field of characteristic zero.

Let K be a perfect field of characteristic p > 0; A1 := K〈x, ∂|∂x−x∂=1〉 be the first Weyl algebra; and Z:=K[X:=xp, Y:=∂p] be its centre. It is proved that (i) the restriction map res : AutK(A1)→ AutK(Z), σ ↦ σ|Z is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z)|(τ)=1}, where (τ) is the Jacobian of τ, (Note that AutK(Z)=K* ⋉ Γ, and if K is not perfect then im(res) ≠ Γ.); (ii) the bijection res : AutK(A1) → Γ is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res−1 is found via differential operators (Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper:

The Skolem–Mahler–Lech theorem states that if $f(n)$ is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that $f(m)$ is equal to 0 is the union of a finite number of arithmetic progressions in $m\ge 0$ and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y, and $\sigma$ is an automorphism of Y, then the set of m such that $\sigma^m({\bf q})$ lies in X is a union of a finite number of complete doubly-infinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem.

The Abhyankar–Sathaye Embedded Hyperplane Problem asks whether any hypersurface of ${{\mathbb{C}}^{n}}$ isomorphic to ${{\mathbb{C}}^{n-1}}$ is rectifiable, i.e., equivalent to a linear hyperplane up to an automorphism of ${{\mathbb{C}}^{n}}$. Generalizing the approach adopted by Kaliman, Vénéreau, and Zaidenberg, which consists in using almost nothing but the acyclicity of ${{\mathbb{C}}^{n-1}}$, we solve this problem for hypersurfaces given by polynomials of $\mathbb{C}\left[ x,y,{{z}_{1}},...,{{z}_{k}} \right]$ as in the title.

Le problème de Jung-Nagata $\left( cf.\,\left[ \text{J} \right],\,\left[ \text{N} \right] \right)$ consiste à savoir s'il existe des automorphismes de $k\left[ x,\,y,\,z \right]$ qui ne sont pas modérés. Nous proposons une approche nouvelle de cette question, fondée sur l'utilisation de la théorie des automates et du polygone de Newton. Cette approche permet notamment de généraliser de façon significative les résultats de $\left[ \text{A} \right]$.