‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

We consider the Zariski space of all places of an algebraic function field F |K of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zerodimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of field...

Let G be the group of rational points of a connected reductive p-adic group and let K be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on G are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of K. In this paper we show that spaces of generalized spherical fu...

The result of [6] is the existence of an infinite family of Zariski dense surface subgroups of fixed genus inside SL(3,Z); here we exhibit such subgroups inside SL(4,Z) and symplectic groups. In this setting the power of such a result comes in large part from the conclusion that the groups are Zariski dense the existence of surface groups inside SL(4,Z) can be proved fairly easily, since it’s n...

Let X = C. In this paper we present an algorithm that computes the de Rham cohomology groups H dR(U, C) where U is the complement of an arbitrary Zariski-closed set Y in X . Our algorithm is a merger of the algorithm given by T. Oaku and N. Takayama ([7]), who considered the case where Y is a hypersurface, and our methods from [9] for the computation of local cohomology. We further extend the a...

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 study Zariski pairs of sextics which are distinguished by the Alexander polynomials. For this purpose, we present two constructive methods to produce explicit sextics of non-torus type with given configuration of simple singularities.

Let R be a commutative hyperring with identity. We introduceand study prime fuzzy hyperideals of R. We investigate the Zariski topologyon FHspec(R), the spectrum of prime fuzzy hyperideals of R.