Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a refreshing view on this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in R in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. ...

Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-map of smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchi and Silverman conjectured if $P \in X({\overline{\mathbf Q}})$ point with well-defined forward orbit, then the growth rate height along orbit exists, coincides first dynamical degree $\lambda_1(f)$ $f$ $P$ Zariski dense in $X$. In this note,...

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime p for the reduction modulo p of an indecomposable polynomial P (x) ∈ Z[x] to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if f(t1, . . . , tr, x) is an indecomposable polynomial in several variables with coefficie...

This terminology is explained by Bogomolov decomposition theorem which states that, up to a finite étale covering, each holomorphic symplectic manifold is a product of several irreducible ones and a torus. Let X be a holomorphic symplectic manifold with a holomorphic symplectic form ω. Let D be a smooth divisor on X. At each point of D, the restriction of ω to D has one-dimensional kernel. This...

3 Topics in Commutative Algebra 2 3.1 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Quotient Rings and Homomorphisms . . . . . . . . . . . . . . 5 3.4 The Characteristic of a Ring . . . . . . . . . . . . . . . . . . . 7 3.5 Polynomial Rings in Several Variables . . . . . . . . . . . . . . 7 3.6...

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of ...

We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with (N,+,×) if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in Z. Notably, by constructing a nontrivial...

In [2], Billera proved that the R-algebra of continuous piecewise polynomial functions (C0 splines) on a d-dimensional simplicial complex 1 embedded in Rd is a quotient of the Stanley–Reisner ring A1 of 1. We derive a criterion to determine which elements of the Stanley–Reisner ring correspond to splines of higher-order smoothness. In [5], Lau and Stiller point out that the dimension of C k (1)...

then V is a primitive Fano variety of dimensionM , that is, Pic V = ZKV and (−KV ) is ample. The purpose of this note is to sketch a proof of the following Theorem 1. A general (in the sense of Zariski topology) variety V is birationally superrigid. In particular, V admits no non-trivial structures of a rationally connected fibration, any birational map V 99K V ♯ onto a Fano variety with Q-fact...

A series of Zariski pairs and four Zariski triplets were found by using lattice theory of K3 surfaces. There is a Zariski triplet of which one member is a deformation of another.