‎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.

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...

Modern model theory can be viewed as a subject obsessed with notions of dimension, with the key examples furnished by linear dimension on the one hand, and the dimension of an algebraic variety (or, from another point of view, transcendences degree) on the other. There are several rigorous, and not always equivalent, notions of abstract dimension in use. For historical reasons the one we use is...

In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the embedded topology a line arrangement which generalizes $$\mathcal {I}$$ -invariant introduced by Artal, Florens and author. This new is called loop-linking number in present paper. We refine result proving that homeomorphism type complement. give two effective methods to compute this invariant, both are based on braid monodro...

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.