Matematicheskii Sbornik

The maximum possible values of the Jordan constant of the automorphism group of a smooth rational two-dimensional quadric over a field of characteristic zero are found in their dependence on the arithmetic properties of the field.

An example of an almost commutative, almost Engel and almost finitely based $L$-variety is presented that is generated by a multiplicative vector space distinct from its enveloping algebra. Also, an example of a critical multiplicative vector space is constructed that is not a linear algebra.

Let $A=\mathbb{F}[x_1,…,x_n]/(f_1,…,f_n)$ be a graded complete intersection Artinian algebra where $\mathbb{F}$ is a field of characteristic zero. The grading on $A$ induces a natural grading on $\operatorname{Der}_{\mathbb{F}}(A)$. Halperin proposed a famous conjecture: $\operatorname{Der}_{\mathbb{F}}(A)_{<0}=0$, which implies the collapsing of the Serre spectral sequence for an orientable fibration with fibre an elliptic space with no cohomology in odd degrees. In the context of singularity theory the second author proposed the same conjecture in the special case when $f_i=\partial f/\partial x_i$ for a single polynomial $f$.
H. Chen, the second author and Zuo [5] proved Halperin's conjecture assuming that the degrees of the $f_i$ are bounded below by a constant depending on the number $n$ of variables and the degrees of variables. In this paper, in the special case when $f_i=\partial f/\partial x_i$ for a single polynomial $f$, we refine their result by giving a better bound, which is independent of $n$.