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.

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 URS AND URSIMS FOR P-ADIC MEROMORPHIC FUNCTIONS INSIDE A DISC

2001 ◽  
Vol 44 (3) ◽  
pp. 485-504 ◽  
Author(s):  
Abdelbaki Boutabaa ◽  
Alain Escassut
Keyword(s):  
Characteristic Function ◽  
Analytic Functions ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Closed Field ◽  
Open Disc ◽  
Absolute Value ◽  
Unique Range Set ◽  
Bounded Characteristic ◽  
Primary 12H25

AbstractLet $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

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.

scholarly journals The Ultrametric Corona Problem and spherically complete fields

2010 ◽  
Vol 53 (2) ◽  
pp. 353-371 ◽  
Author(s):  
Alain Escassut
Keyword(s):  
Maximal Ideal ◽  
Analytic Functions ◽  
Unit Disc ◽  
Unique Continuation ◽  
Open Unit ◽  
Closed Field ◽  
Corona Problem ◽  
Open Unit Disc ◽  
Bounded Analytic Functions

AbstractLet K be a complete ultrametric algebraically closed field and let A be the Banach K-algebra of bounded analytic functions in the ‘open’ unit disc D of K provided with the Gauss norm. Let Mult(A,‖ · ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Multm(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal and let Multa(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. We complete the characterization of continuous multiplicative norms of A by proving that the Gauss norm defined on polynomials has a unique continuation to A as a norm: the Gauss norm again. But we find prime closed ideals that are neither maximal nor null. The Corona Problem on A lies in two questions: is Multa(A, ‖ · ‖) dense in Multm(A, ‖ · ‖)? Is it dense in Multm(A, ‖ · ‖)? In a previous paper, Mainetti and Escassut showed that if each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖), then the answer to the first question is affirmative. In particular, the authors showed that when K is strongly valued each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖). Here we prove that this uniqueness also holds when K is spherically complete, and therefore so does the density of Multa(A, ‖ · ‖) in Multm(A, ‖ · ‖).

scholarly journals Subordination Properties of Meromorphic Kummer Function Correlated with Hurwitz–Lerch Zeta-Function

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 192
Author(s):  
Firas Ghanim ◽  
Khalifa Al-Shaqsi ◽  
Maslina Darus ◽  
Hiba Fawzi Al-Janaby
Keyword(s):  
Operator Theory ◽  
Analytic Functions ◽  
Meromorphic Functions ◽  
Special Function ◽  
Function Theory ◽  
Applied Mathematics ◽  
Analytical Characterization ◽  
Kummer Functions ◽  
Lerch Zeta Function

Recently, Special Function Theory (SPFT) and Operator Theory (OPT) have acquired a lot of concern due to their considerable applications in disciplines of pure and applied mathematics. The Hurwitz-Lerch Zeta type functions, as a part of Special Function Theory (SPFT), are significant in developing and providing further new studies. In complex domain, the convolution tool is a salutary technique for systematic analytical characterization of geometric functions. The analytic functions in the punctured unit disk are the so-called meromorphic functions. In this present analysis, a new convolution complex operator defined on meromorphic functions related with the Hurwitz-Lerch Zeta type functions and Kummer functions is considered. Certain sufficient stipulations are stated for several formulas of this defining operator to attain subordination. Indeed, these outcomes are an extension of known outcomes of starlikeness, convexity, and close to convexity.

scholarly journals SETS OF RANGE UNIQUENESS IN $p$-ADIC FIELDS

2007 ◽  
Vol 50 (2) ◽  
pp. 263-276 ◽  
Author(s):  
K. Boussaf ◽  
A. Boutabaa ◽  
A. Escassut
Keyword(s):  
Analytic Functions ◽  
Finite Limit ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Absolute Value

AbstractWe study sets of range uniqueness (SRUs) for analytic functions inside a disc of an algebraically closed field $K$ complete with respect to an ultrametric absolute value. The SRUs we obtain are converging sequences. We first obtain results that look like those known in $\mathbb{C}$ but involve a weaker hypothesis than in $\mathbb{C}$: let $(a_n)$ be a sequence of limit $a$ in a disc $d(a,r^-)$ such that $|a_n-a|$ is a strictly decreasing sequence. If the sequence $(a_n)$ does not make an SRU for the set $\mathcal{A}(d(a,r^-))$ of analytic functions inside $d(a,r^-)$, then, for a certain integer $k\in\mathbb{Z}$, the sequence$$ \bigg(\frac{a_{n+k}-a}{a_n-a}\bigg) $$has a finite limit in $K$ and the sequence$$ \bigg(\frac{\log|a_{n+k}-a|}{\log|a_n-a|}\bigg) $$has a finite rational limit. Next, we show that if the sequence$$ \frac{\log(a_{n+1}-a)}{\log(a_n-a)} $$converges to a limit $b\geq1$ in such a way that $-b\log|a_{n}-a|\lt-b\log|a_{n+1}-a|$ and if $\log|a_{n}-a|-b\log|a_{n+1}-a|$ has limit $0$ or $+\infty$ and if $b^k\notin\mathbb{Q}$ whenever $b>1$ and $k\in \mathbb{N}^*$, then the sequence $(a_n)$ is an SRU for $\mathcal{A}(d(a,r^-))$. In particular, for every $\gamma\in\;]0,1[\;\cup\;]1,+\infty[$, $L\in\mathbb{Q}\;\cap\;]0,+\infty[$ and $b\geq 1$, there exist SRUs for $\mathcal{A}(d(a,r^-))$ of the form $\{a_n\mid n\in\mathbb{N}\}$ such that$$ \lim_{n\rightarrow+\infty}\frac{-\log|a_n-a|}{b^nn^\gamma}=L. $$For example, if $\gamma\in\mathbb{N}$ with $\gamma\neq0,1$, there exist SRUs of the form $\{a_n\mid n\in\mathbb{N}\}$ such that $-\log |a_n-a|=Ln^\gamma$ for all $n\in\mathbb{N}^*$. The latter result ceases to hold when $\gamma=1$. Many examples and counterexamples are provided.

A Module-theoretic Characterization of Algebraic Hypersurfaces

2018 ◽  
Vol 61 (1) ◽  
pp. 166-173
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Vector Fields ◽  
Exterior Power ◽  
Algebraic Hypersurfaces ◽  
Theoretic Characterization

AbstractIn this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

Central automorphisms of free nilpotent Lie algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750205
Author(s):  
Özge Öztekin ◽  
Naime Ekici
Keyword(s):  
Lie Algebras ◽  
Finite Rank ◽  
Characteristic Zero ◽  
Sufficient Conditions ◽  
Nilpotency Class ◽  
Nilpotent Lie Algebra ◽  
Central Automorphism ◽  
Nilpotent Lie Algebras ◽  
Central Automorphisms

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

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.

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