Related papers: Isomorphism classes and stably isomorphisms of dou…

In this paper, we study the double Danielewski varieties which arose from the research on the classical Cancellation Problem. We describe the Makar-Limanov invariant and locally nilpotent derivations of these varieties. And in a subsequent…

We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms…

The cancellation problem asks if two complex algebraic varieties X and Y of the same dimension such that X\times\mathbb{C} and Y\times\mathbb{C} are isomorphic are isomorphic. Iitaka and Fujita established that the answer is positive for a…

In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the…

In this paper we extend a result of Dubouloz on the Cancellation Problem in higher dimensions ($\geqslant 2$) over the field of complex numbers to fields of arbitrary characteristic. We then apply the generalised result to describe the…

Let $k$ be a field. In this paper, we consider Double Danielewski type algebras over an affine factorial $k$-domain $R$. We observe that this family produces a non-cancellative family of algebras over $R$. Further, when $k$ is a field of…

We address a variant of Zariski Cancellation Problem, asking whether two varieties which become isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property is easily checked for…

Let $K$ be an arbitrary field of characteristic 0, and $\Aff^n$ the $n$-dimensional affine space over $K$. A well-known cancellation problem asks, given two algebraic varieties $V_1, V_2 \subseteq \Aff^n$ with isomorphic cylinders $V_1…

In this note we rectify the proof of Theorem 3.11 in [arXiv:2403.02876]. We also present a set of examples at the end discussing various cases.

We relate the graph isomorphism problem to the solvability of certain systems of linear equations with nonnegative variables. This version replaces the two previous versions of this paper.

This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski…

We give a direct proof of a cancellation formula raised in [7] on the level of differential forms. We also obtain more cancellation formulas for even dimensional Riemannian manifolds with a complex line bundle involved. Relations among…

Let $X$ and $X'$ be affine algebraic varieties over a field $\mathbb{k}$. The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ implies $X\cong X'$. In…

In this article, we discuss some recent developments of the Zariski Cancellation Problem in the setting of noncommutative algebras and Poisson algebras.

Consider two planar graphs which are subject to edge insertions and deletions. We show that whether the two graphs are isomorphic can be maintained with first-order logic formulas and auxiliary data of polynomial size. This places the…

The Zariski cancellation problem plays a central role in affine algebraic geometry and noncommutative algebra, with locally nilpotent derivations providing a fundamental invariant-theoretic approach. This article presents a unified survey…

In this paper, over a field of positive characteristic we exhibit an infinite family of counter examples to the Zariski Cancellation Problem in higher dimensions ($\geqslant 3$) which are pairwise non-isomorphic and also non-isomorphic to…

The Danielewski hypersurfaces are the hypersurfaces $X_{Q,n}$ in $\mathbb{C}^3$ defined by an equation of the form $x^ny=Q(x,z)$ where $n\geq1$ and $Q(x,z)$ is a polynomial such that $Q(0,z)$ is of degree at least two. They were studied by…

We construct all possible noncommutative deformations of a Kleinian singularity ${\mathbb C}^2/\Gamma$ of type $D_n$ in terms of generators and relations, and solve the problem of when two deformations are isomorphic. We prove that all…

We study the Zariski cancellation problem for Poisson algebras in three variables. In particular, we prove those with Poisson bracket either being quadratic or derived from a Lie algebra are cancellative. We also use various Poisson algebra…