In this paper, by using a special family of filters $mathcal{F}$ on an EQ-algebra $E$, we construct a topology $mathcal{T}_{mathcal{mathcal{F}}}$ on $E$ and show that $(E,mathcal{T}_{mathcal{F}})$ is a topological EQ-algebra. First of all, we give some properties of topological EQ-algebras and investigate the interaction of topological EQ-algebras and quotient topological EQ-algebras. Then we o...

The notion of strict closedness rings was given by J. Lipman [Amer. Math. 93 (1971), pp. 649–685] in connection with a conjecture O. Zariski. present purpose is to give practical method construction strictly closed rings. It also shown that the Stanley-Reisner simplicial complexes (resp. $F$-pure satisfying condition $(\operatorname {S}_2)$ Serre) are weakly Arf)

Suppose C is an algebraic curve, f is a rational function on C defined over Q, and A is a fractional ideal of Q. If f is not equivalent to a polynomial, then Siegel’s theorem gives a necessary condition for the set C(Q)∩f−1(A) to be infinite: C is of genus 0 and the fiber f−1(∞) consists of two conjugate quadratic real points. We consider a converse. Let P be a parameter space for a smooth fami...

In this note we show that the family of Cayley graphs a finitely generated subgroup ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is expanders under certain algebraic conditions. Here more precise formulation our main result. For positive integer $c_0$, say polynomial $c_0$-admissible if degree irreducible factors $f$ are distinct integers with prime at least ...

In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with arbitrary l poles of prescribed orders, such that for every geometric point f in U(Q), the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to a...

where we view Gm(S) as the subgroup of invertible scalars in GLn(S). Define PGLn = s ◦ P̃GLn to be its sheafification (taken in the sense of Problem 8 on PS 2). Our goal will be to show that PGLn is an affine group scheme that represents automorphisms of projective space in the sense that PGLn(S) ' AutS(S × Pn−1). Before we begin, we require some preliminaries about Zariski sheaves. Let F : Sch→...

Toric varieties are a special class of rational varieties defined by equations of the form monomial = monomial. For a good brief survey of the history and role of toric varieties see [10]. Any toric variety X contains a cover by affine open sets described in terms of arrangements (called fans) of convex bodies in Rr. The coordinate rings of each of these affine open sets is a graded ring genera...

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