Related papers: Rees valuations

We characterize the diacriticals of special pencils. We also initiate higher dimensional dicritical theory.

We discuss the work of Abhyankar on dicritical divisors with a special focus on the algebraic aspects of this work. We also discuss related work on local quadratic transforms, infinitely near points and Rees valuation rings of an ideal.

We study the shifted convolution sum of the divisor function and some other arithmetic functions.

In this paper, we explain how some basic facts about valuation can help clarify many questions about divisibility in integral domains.

In [AL11], S.S Abhyankar and I. Luengo introduce a new theory of dicritical divisors in the most general framework. Here we simplify and generalize their results (see Theorems 3.1 and 3.2).

Our paper is devoted to several problems from the field of modified divisors: namely exponential and infinitary divisors. We study the behaviour of modified divisors, sum-of-divisors and totient functions. Main results concern with the…

In geometric terms, given a singular foliation of the plane, a dicritical divisor is (whenever it exists) an irreducible component of the exceptional divisor which is transverse to the foliation. Abhyankar gave recently a definition of the…

We study the average order of the divisor function, as it ranges over the values of binary quartic forms that are reducible over the rationals.

We study the value-distribution of Dirichlet polynomials on the critical line $\Re(s)=\tfrac{1}{2}$. As a consequence, we prove a corollary on small consecutive gaps between zeros of the Riemann zeta function. We also examine the…

In this article, we study the distribution of values of Dirichlet $L$-functions, the distribution of values of the random models for Dirichlet $L$-functions, and the discrepancy between these two kinds of distributions. For each question,…

We give a criterion for a real divisor to be rational and semiample.

A class of e-variables is introduced and analyzed. Some examples are presented.

In this paper we present a divisorial valuation with irrational volume using an algebro-geometric construction.

We use reproducing kernel methods to study various rigidity problems. The methods and setting allow us to also consider the non-positive case.

We investigate the existence and multiplicity of solutions for higher order discrete boundary value problems via critical point theory.

Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.

This paper shows a finiteness property of a divisorial valuation in terms of arcs. First we show that every divisorial valuation over an algebraic variety corresponds to an irreducible closed subset of the arc space. Then we define the…

In this paper we show that if R is a discrete valuation ring, then R is a filtered ring. We prove some properties and relation when R is a discrete valuation ring.

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by Bordell\'{e}s \cite{Bo} and Liu-Wu-Yang \cite{LWY}.

We investigate the average order of the divisor function at values of totally reducible binary cubic forms and discuss some applications.