ZEROS OF ULTRAMETRIC MEROMORPHIC FUNCTIONS f′ fn(f − a)k − α

2008 ◽  
Vol 01 (03) ◽  
pp. 415-429 ◽  
Author(s):  
Jacqueline Ojeda
Keyword(s):  
Meromorphic Function ◽  
Characteristic Zero ◽  
Meromorphic Functions ◽  
Sufficient Conditions ◽  
Small Function ◽  
Closed Field ◽  
Open Disk ◽  
Algebraically Closed Field ◽  
Absolute Value ◽  
Special Value

Let 𝕂 be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. Similarly to the Hayman problem, here we study meromorphic functions in 𝕂 or in an open disk that are of the form f′ fn(f − a)k − α with α a small function, in order to find sufficient conditions on n, k assuring that they have infinitely many zeros. We first define and characterize a special value for a meromorphic function and check that, if it exists, it is unique. So, such values generalize Picard exceptional values.

scholarly journals URSCM OR BI-URSCM FOR p-ADIC ANALYTIC OR MEROMORPHIC FUNCTIONS INSIDE A DISK

2007 ◽  
Vol 49 (1) ◽  
pp. 121-126
Author(s):  
ABDELBAKI BOUTABAA ◽  
ALAIN ESCASSUT
Keyword(s):  
General Result ◽  
Analytic Functions ◽  
Characteristic Zero ◽  
Meromorphic Functions ◽  
Closed Field ◽  
Open Disk ◽  
Second Main Theorem ◽  
Absolute Value ◽  
Small Functions

Abstract.Let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value. In a previous paper, we had found URSCM of 7 points for the whole set of unbounded analytic functions inside an open disk. Here we show the existence of URSCM of 5 points for the same set of functions. We notice a characterization of BI-URSCM of 4 points (and infinity) for meromorphic functions in K and can find BI-URSCM for unbounded meromorphic functions with 9 points (and infinity). The method is based on the p-Adic Nevanlinna Second Main Theorem on 3 Small Functions applied to unbounded analytic and meromorphic functions inside an open disk and we show a more general result based upon the hypothesis of a finite symmetric difference on sets of zeros, counting multiplicities.

scholarly journals HEIGHTS OF FUNCTION FIELD POINTS ON CURVES GIVEN BY EQUATIONS WITH SEPARATED VARIABLES

2012 ◽  
Vol 23 (09) ◽  
pp. 1250089
Author(s):  
TA THI HOAI AN ◽  
NGUYEN THI NGOC DIEP
Keyword(s):  
Characteristic Zero ◽  
Function Field ◽  
Sufficient Conditions ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Separated Variables

Let P and Q be polynomials in one variable over an algebraically closed field k of characteristic zero. Let f and g be elements of a function field K over k such that P(f) = Q(g). We give conditions on P and Q such that the height of f and g can be effectively bounded, and moreover, we give sufficient conditions on P and Q under which f and g must be constant.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

Abelian Gradings on Upper Block Triangular Matrices

2012 ◽  
Vol 55 (1) ◽  
pp. 208-213 ◽  
Author(s):  
Angela Valenti ◽  
Mikhail Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Block Triangular Matrix

AbstractLet G be an arbitrary finite abelian group. We describe all possible G-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

scholarly journals Corrections to “Seminormal rings and weakly normal varieties”

1987 ◽  
Vol 107 ◽  
pp. 147-157 ◽  
Author(s):  
Marie A. Vitulli
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Regular Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Regular Functions ◽  
Normal Varieties

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

scholarly journals The equation y′ = fy in zero residue characteristic

1991 ◽  
Vol 33 (2) ◽  
pp. 149-153
Author(s):  
Alain Escassut ◽  
Marie-Claude Sarmant
Keyword(s):  
Uniform Convergence ◽  
Characteristic Zero ◽  
Residue Class ◽  
Rational Functions ◽  
Closed Field ◽  
Class Field ◽  
Absolute Value ◽  
Residue Class Field ◽  
Analytic Elements ◽  
Residue Characteristic

Let K be an algebraically closed field complete with respect to an ultrametric absolute value |.| and let k be its residue class field. We assume k to have characteristic zero (hence K has characteristic zero too).Let D be a clopen bounded infraconnected set [3] in K, let R(D) be the algebra of the rational functions with no pole in D, let ‖.‖D be the norm of uniform convergence on D defined on R(D), and let H(D) be the algebra of the analytic elements on D i.e. the completion of R(D) for the norm ‖.‖D.

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