Related papers: Zero Order Estimates for Analytic Functions
We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of…
We consider the problem of estimating the multiplicity of a polynomial when restricted to the smooth analytic trajectory of a (possibly singular) polynomial vector field at a given point or points, under an assumption known as the…
The problem of estimating the multiplicity of the zero of a polynomial when restricted to the trajectory of a non-singular polynomial vector field, at one or several points, has been considered by authors in several different fields. The…
We develop a theory of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence, which also includes the possibility of dealing with several systems associated with sufficiently independent…
In a recent work [3], the authors established new results about general linear Mahler systems in several variables from the perspective of transcendental number theory, such as a multivariate extension of Nishioka's theorem. Working with…
We give an upper bound for the zero order of the difference between a Mahler function and an algebraic function. This complements estimates of Nesterenko, Nishioka, and T\"opfer, among others, who considered polynomials evaluated at Mahler…
We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are…
The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a…
The main purpose of this article is to provide new results on algebraic independence of values of Mahler functions and their generalizations. Simultaneously, we establish new measures of algebraic independence for these values. Among the…
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…
This is the first part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. We prove two main results concerning systems that are regular singular at…
In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition…
The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces…
We give some new results on algebraic independence within Mahler's method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In…
We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-$*$ closure, much in the spirit of the corresponding…
In recent years, a multiplier ideal defined on arbitrary varieties, so called Mather multiplier ideal, has been developed independently by Ein-Ishii-Mustata, and de Fernex-Docampo. With this new tool, we have a chance of extending some…
We provide a generalization of first-order necessary conditions of optimality for infinite-dimensional optimization problems with a finite number of inequality constraints and with a finite number of inequality and equality constraints. Our…
The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via…
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…
Multiplier ideals, and the vanishing theorems they satisfy, have found many applications in recent years. In the global setting they have been used to study pluricanonical and other linear series on a projective variety. More recently, they…