Given an algebra B over a field k, we study conditions under which a Lie subalgebra of Der(B) is locally finite as a set of derivations. As an application of our results, we show that if X is a quasi-affine variety over an arbitrary field k, and if L is a finitely generated solvable Lie subalgebra of Der O(X) consisting of locally finite derivations, then L is locally finite. If, moreover, k is algebraically closed and of characteristic zero, and X is irreducible and affine, then L is integrable.

Given an algebra B B over a field k {\mathrm {\mathbf {k}}} , we study conditions under which a Lie subalgebra of D e r k ( B ) Der_{\mathrm {\mathbf {k}}}(B) is locally finite as a set of derivations. As an application of our results, we show that if X X is a quasi-affine variety over an arbitrary field k {\mathrm {\mathbf {k}}} , and if L \mathscr {L} is a finitely generated solvable Lie subalgebra of D e r k O X ( X ) Der_{\mathrm {\mathbf {k}}}\mathscr {O}_X(X) consisting of locally finite derivations, then L \mathscr {L} is locally finite. If, moreover, k {\mathrm {\mathbf {k}}} is algebraically closed and of characteristic zero, and X X is irreducible and affine, then L \mathscr {L} is integrable.

Let k^[6] denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL _2(k) on k^[6] induced by the irreducible representation of SL _2 of degree 5 (the binary quintic representation). We consider the ring Q = (k^[6])^ SL _2 of invariant polynomials and show that Aut_k(Q) = k^* , where Aut_k(Q) is the group of k-algebra automorphisms of Q. Based on this result, we show that the group of SL _2 -equivariant polynomial automorphisms of k^[6] is isomorphic to k^* .

We study a class of combinatorial objects that we call “decorated trees”. These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and the decorations are required to satisfy certain conditions. The class of decorated trees includes different types of trees used in algebraic geometry, such as the Eisenbud and Neumann diagrams for links of singularities and the Neumann diagrams for links at infinity of algebraic plane curves. By purely combinatorial means, we recover some formulas that were previously understood to be “topological”. In this way, we extend the generality of those formulas and show that they are in fact “combinatorial”.

A ring R is said to be rigid if the only locally nilpotent derivation of R is the zero derivation. Let G be an abelian group, and B = (direct sum of B_i for i in G) be a G-graded commutative integral domain of characteristic 0. For each subgroup H of G, consider the Veronese subring B(H) of B, defined by B(H) = (direct sum of the B_i for i in H). We study the following questions. If B is non-rigid, does it follow that B(H) is non-rigid? Can derivations of B(H) be extended to derivations of B? What are the properties of the set of subgroups H of G such that B(H) is non-rigid?

We give several criteria for a ring to be a UFD, including generalizations of some criteria due to P. Samuel. These criteria are applied to construct, for any field k, (1) a Z-graded non-noetherian rational UFD of dimension 3 over k, and (2) k-affine rational UFDs defined by trinomial relations.

Let B be a commutative ℤ -graded domain of characteristic zero. An element f of B is said to be cylindrical if it is nonzero, homogeneous of nonzero degree, and such that B ( f ) is a polynomial ring in one variable over a subring. We study the relation between the existence of a cylindrical element of B and the existence of a nonzero locally nilpotent derivation of B . Also, given d ≥ 1, we give sufficient conditions that guarantee that every derivation of B^(d) = ⊕_i ∈ℤ B_di can be extended to a derivation of B . We generalize some results of Kishimoto, Prokhorov and Zaidenberg that relate the cylindricity of a polarized projective variety ( Y , H ) to the existence of a nontrivial G a -action on the affine cone over ( Y , H ).

We prove Freudenburg's Freeness Conjecture: Let B be the polynomial ring in three variables over a field of characteristic zero, let D : B --> B be a nonzero locally nilpotent derivation, and let A = ker(D). Then B is a free A-module, and there exists a basis $(e_i)_{i \in \mathbb{N}}$ of B such that deg$_D(e_i) = i$ for all $i \in \mathbb{N}$.

Let B be an algebra over a field k and let Der(B) be the set of k-derivations from B to B. We define what it means for a subset of Der(B) to be a locally nilpotent set. We prove some basic results about that notion and explore the following questions. Let L be a Lie subalgebra of Der(B); if every element of L is a locally nilpotent derivation then does it follow that L is a locally nilpotent set? Does it follow that L is a nilpotent Lie algebra?

Fix a field k of characteristic zero. If a1,…,an (n≥3) are positive integers, the integral domainBa1,…,an=k[X1,…,Xn]/〈X1a1+⋯+Xnan〉 is called a Pham-Brieskorn ring. It is conjectured that if ai≥2 for all i and ai=2 for at most one i, then Ba1,…,an is rigid. (A ring B is said to be rigid if the only locally nilpotent derivation D:B→B is the zero derivation.) The conjecture is known to be true when n=3, and in certain special cases when n≥4. This article settles several cases not covered by previous results. For instance, we show that if a≥n≥4 then Ba,…,a is rigid (where ‘a’ occurs n times), and that if ∑i=1n1ai≤1n−2 then Ba1,…,an is stably rigid.

Let k be a field of characteristic zero and B a commutative integral domain that is also a finitely generated k-algebra. It is well known that if k is algebraically closed and the “field Makar-Limanov” invariant FML(B) is equal to k, then B is unirational over k. This article shows that, when k is not assumed to be algebraically closed, the condition FML(B) = k implies that there exists a nonempty Zariski-open subset U of Spec B with the following property: for each prime ideal $$ \mathfrak{p} $$ ∈ U, the κ($$ \mathfrak{p} $$)-algebra κ($$ \mathfrak{p} $$)⊗kB can be embedded in a polynomial ring in n variables over κ($$ \mathfrak{p} $$), where n = dim B and κ($$ \mathfrak{p} $$) = $$ {B}_{\mathfrak{p}}/\mathfrak{p}{B}_{\mathfrak{p}} $$.

Let A be a geometrically integral algebra over a field k. We prove that, for any affine k-domain R, if there exists an extension field K of k such that R subset of K circle times(k) A and R not subset of K, then there exists an extension field L of k such that R subset of L circle times(k) A and trdeg(k)(L) < trdeg(k)(R). This generalizes a result of Freudenburg, namely, the fact that this is true for A = k([1]).

We investigate the structure of commutative integral domains B of characteristic zero by studying the kernels of locally nilpotent derivations D:B→B.

Among the most important tools in the study of locally nilpotent derivations (LNDs) of commutative rings are its Z-gradings and the homogeneous derivations associated to them. Gradings which involve other totally ordered abelian groups have also been used to study LNDs, though to a lesser extent. Gradings which involve non-totally ordered abelian groups have been largely ignored in this context, since it is no longer possible to associate a highest-degree homogeneous derivation to a given derivation in this case. However, it turns out that one can still get valuable information about LNDs from such gradings. In [7], the second and third authors studied LNDs of certain rings graded by a finite cyclic group. Their results were applied to show that some families of Pham-Brieskorn threefolds are rigid, i.e., their coordinate rings have no nonzero LNDs. In the present work, we generalize the theory developed in that paper to the case of rings graded by arbitrary abelian groups. Let B be a domain of characteristic zero graded by an abelian group G. For any subgroup H of G, let BH be the subring of B generated by the nonzero homogeneous elements of B whose degrees belong to H. An element x of B is G-critical if it is homogeneous and nonzero, and if there exists a subgroup H of G such that deg x / ∈ H and B = BH [x]. Our main result is Theorem 6.2, which states that Dx = 0 whenever D is a homogeneous LND of B and x ∈ B is a G-critical element. Moreover, if y is a second G-critical element and xB 6= yB, then either Dx = 0 or Dy = 0. As an application of this theorem, we settle some of the cases of Pham-Brieskorn threefolds which were out of the reach of the earlier paper. As a second application, we give a short proof of the fact that the Derksen invariant of Russell’s cubic threefold is nontrivial. We assume throughout that rings are commutative with identity. Given the ring B, the units of B are denoted by B∗. Given the integer n ≥ 0, the polynomial ring in n variables over B is denoted B. If B is a domain, the field of fractions of B is denoted frac(B). The cyclic group of order n is indicated by Zn.

Let f:ℂ^2 →ℂ be a polynomial map. Let ℂ^2 ⊂ X be a compactification of ℂ^2 where X is a smooth rational compact surface and such that there exists a morphism of varieties Φ :X→ℙ^1 which extends f. Put 𝒟=X∖ℂ^2; 𝒟 is a curve whose irreducible components are smooth rational compact curves and all its singularities are ordinary double points. The dual graph of 𝒟 is a tree. We are interested in this tree, and we analyse its complexity in terms of the genus of the generic fiber of f.

We determine the Newton trees of the rational polynomials of simple type, thus filling a gap in the proof of the classification of these polynomials given by Neumann and Norbury.

Let k be a field. We study infinite strictly descending sequences A0⊃A1⊃⋯ of rings where each Ai is a polynomial ring in two variables over k, the aim being to describe those sequences satisfying ⋂i=0∞Ai≠k. We give a complete answer in characteristic zero, and partial results in arbitrary characteristic. We apply those results to the study of dominant morphisms A2→An and their factorizations, where n∈{1,2} and An is the affine n-space over k.

Let k be an algebraically closed field. A polynomial F in k[X,Y] is said to be "generally rational" if, for almost all c in k, the curve " F= c '' is rational. It is well known that, if char(k)=0, F is generally rational iff there exists G in k(X,Y) such that k(F,G)=k(X,Y). We give analogous results valid in arbitrary characteristic.

This paper is a survey of two subjects: the first part is devoted to field generators in two variables, and the second to birational endomorphisms of the affine plane. Each one of these subjects originated in Abhyankar's seminar in Purdue University in the 1970s. Note that the part on field generators is more than a survey, since it contains a considerable amount of new material.

Let k be a field. A "field generator" is a polynomial F in k[X,Y] satisfying k(F,G) = k(X,Y) for some G in k(X,Y). If G can be chosen in k[X,Y], we call F a "good field generator"; otherwise, F is a "bad field generator". These notions were first studied by Abhyankar, Jan and Russell in the 1970s. The present paper introduces and studies the notions of "very good" and "very bad" field generators. We give theoretical results as well as new examples of bad and very bad field generators.