On valued function fields II. Regular functions and elements with the uniqueness property.
On valued function fields II. Regular functions and elements with the uniqueness property.
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关于有价值的函数域 II.
DOI：
10.1515/crll.1990.412.128
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
M. Matignon
中科院分区：
文献类型：
--
作者：
F. Pop;B. Green;M. Matignon
The purpose of this work is to continue a study of valued function fields of one variable over a constant field, that is function fields of one variable which are equipped with an arbitrary valuation on the constant field and a Prolongation to the function field, such that the corresponding residue fields again give a function field in one variable. The first part of the work, [G-M-P], was concerned with the defect of a valued function field and an inequality relating the genus of a function field in one variable over a constant field and equipped with finitely many valuations which coincide on the base field, to the genera of the corresponding residual function fields. In this paper we shall define a divisor reduction map and show the existence of elements with the uniqueness property over an arbitrary henselian base field. By an element with the uniqueness property for a valued function field we mean an element such that the valuation on the function field is the unique Prolongation of the functional (Gaussian) valuation on the rational function field obtained by adjoining this element to the constant field. More specially, when the unique Prolongation is unramified and defectless this element with the uniqueness property is called a regulär function for the valued function field.
The purpose of this work is to continue a study of valued function fields of one variable over a constant field, that is function fields of one variable which are equipped with an arbitrary valuation on the constant field and a Prolongation to the function field, such that the corresponding residue fields again give a function field in one variable. The first part of the work, [G-M-P], was concerned with the defect of a valued function field and an inequality relating the genus of a function field in one variable over a constant field and equipped with finitely many valuations which coincide on the base field, to the genera of the corresponding residual function fields. In this paper we shall define a divisor reduction map and show the existence of elements with the uniqueness property over an arbitrary henselian base field. By an element with the uniqueness property for a valued function field we mean an element such that the valuation on the function field is the unique Prolongation of the functional (Gaussian) valuation on the rational function field obtained by adjoining this element to the constant field. More specially, when the unique Prolongation is unramified and defectless this element with the uniqueness property is called a regulär function for the valued function field.