scholarly journals Taylor coefficients of Anderson generating functions and Drinfeld torsion extensions

2021 ◽  
pp. 1-18
Author(s):  
A. Maurischat ◽  
R. Perkins
Keyword(s):  
Galois Group ◽  
Generating Functions ◽  
Prime Power ◽  
Direct Proof ◽  
Galois Representation ◽  
Roots Of Unity ◽  
Taylor Coefficients ◽  
Tate Module ◽  
Anderson Generating Functions ◽  
Adic Case

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the [Formula: see text]-adic Tate module lies in the [Formula: see text]-adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the [Formula: see text]-adic case.

scholarly journals Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

2011 ◽  
Vol 203 ◽  
pp. 47-100 ◽  
Author(s):  
Yuichiro Hoshi
Keyword(s):  
Isomorphism Class ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Tate Module ◽  
Hyperbolic Curve ◽  
Theoretic Characterization

AbstractLet l be a prime number. In this paper, we prove that the isomorphism class of an l-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-l outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

scholarly journals Pattern Avoidance in Matchings and Partitions

10.37236/2976 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Bijective Proof ◽  
Rook Placements ◽  
Wilf Equivalence ◽  
Diagram Representation

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of Bóna for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

scholarly journals Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

2011 ◽  
Vol 203 ◽  
pp. 47-100 ◽  
Author(s):  
Yuichiro Hoshi
Keyword(s):  
Isomorphism Class ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Tate Module ◽  
Hyperbolic Curve ◽  
Theoretic Characterization

AbstractLetlbe a prime number. In this paper, we prove that the isomorphism class of anl-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-louter Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

scholarly journals GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

2009 ◽  
Vol 51 (2) ◽  
pp. 289-299 ◽  
Author(s):  
ANTOINE D. COSTE ◽  
GARETH A. JONES ◽  
MANFRED STREIT ◽  
JÜRGEN WOLFART
Keyword(s):  
Galois Group ◽  
Riemann Surfaces ◽  
Prime Power ◽  
Algebraic Curves ◽  
Complete Bipartite Graph ◽  
Number Fields ◽  
Absolute Galois Group ◽  
Graph Theoretic ◽  
Galois Orbits ◽  
Fermat Curves

AbstractWe consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods.

Mappings on Decomposable Combinatorial Structures: Analytic Approach

2002 ◽  
Vol 11 (1) ◽  
pp. 61-78 ◽  
Author(s):  
E. MANSTAVIČIUS
Keyword(s):  
Number Theory ◽  
Generating Functions ◽  
Multiplicative Functions ◽  
Combinatorial Structures ◽  
Mean Values ◽  
Taylor Coefficients ◽  
Probabilistic Number ◽  
Generating Series ◽  
Euler Products ◽  
Complex Valued

On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.

PERFORMANCE ANALYSIS OF FREQUENCY-HOPPED SPREAD-SPECTRUM PACKET RADIO NETWORKS PART II: UNSLOTTED MULTIPLE-ACCESS

1996 ◽  
Vol 06 (05) ◽  
pp. 453-474
Author(s):  
KHAIRI ASHOUR MOHAMED ◽  
LÁSZLÓ PAP
Keyword(s):  
Performance Analysis ◽  
Generating Functions ◽  
Spread Spectrum ◽  
Multiple Access ◽  
Joint Probability ◽  
Direct Proof ◽  
Radio Networks ◽  
Packet Radio Networks ◽  
Packet Radio ◽  
Packet Arrival

This paper is concerned with the performance analysis of unslotted frequency-hopped spread spectrum packet radio networks with Reed-Solomon forward error control coding. The following different network alternatives are analyzed in a unified manner (i) random/one-coincidence frequency hopping patterns, (ii) one/several code symbols per dwell interval, (iii) independent/dependent packet arrival processes, (iv) fixed/variable length packets. Joint probability generating functions for the pair-wise hit patterns and the effective multiple-access interference vector are determined for the different alternatives. Closed form formulae and expressions for packet capture probabilities are readily obtained from these generating functions. A direct proof is given that in the case of single symbol/dwell packets symbol errors, are asymptotically independent. Numerical results indicate that the unslotted networks outperform the corresponding slotted ones at low to moderate traffic levels, and the difference between the two networks is diminishing when the number of frequency bins is large.

scholarly journals Generating Functions and Generalized Dedekind Sums

10.37236/1326 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Ira Gessel
Keyword(s):  
Rational Function ◽  
Explicit Formula ◽  
Simple Form ◽  
Generating Functions ◽  
Dedekind Sums ◽  
Roots Of Unity ◽  
Partial Fractions

We study sums of the form $\sum_\zeta R(\zeta)$, where $R$ is a rational function and the sum is over all $n$th roots of unity $\zeta$ (often with $\zeta =1$ excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for $\prod_\zeta (1-xR(\zeta))$. Multisection can be used to evaluate some simple, but important sums. Finally, the method of partial fractions reduces the evaluation of arbitrary generalized Dedekind sums to those of a very simple form.

scholarly journals Developments in the Khintchine-Meinardus Probabilistic Method for Asymptotic Enumeration

10.37236/4581 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Boris L. Granovsky ◽  
Dudley Stark
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Probabilistic Method ◽  
Asymptotic Enumeration ◽  
Combinatorial Objects ◽  
Taylor Coefficients ◽  
Gentile Statistics ◽  
Novel Applications

A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where $S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus' theorem to find the asymptotics of the Taylor coefficients of generating functions of the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and general $S(z)$. We also reformulate the hypotheses of the theorem in terms of the above generating functions. This allows novel applications of the method. In particular, we prove rigorously the asymptotics of Gentile statistics and derive the asymptotics of combinatorial objects with distinct components.

scholarly journals Stirling Numbers of Forests and Cycles

10.37236/3170 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
David Galvin ◽  
Do Trong Thanh
Keyword(s):  
Generating Functions ◽  
Direct Proof ◽  
Random Variable ◽  
Stirling Numbers ◽  
Independent Sets ◽  
Stirling Number ◽  
Asymptotically Normal ◽  
Standard Normal ◽  
Vertex Set ◽  
Number Of Classes

For a graph $G$ and a positive integer $k$, the graphical Stirling number $S(G,k)$ is the number of partitions of the vertex set of $G$ into $k$ non-empty independent sets. Equivalently it is the number of proper colorings of $G$ that use exactly $k$ colors, with two colorings identified if they differ only on the names of the colors. If $G$ is the empty graph on $n$ vertices then $S(G,k)$ reduces to  $S(n,k)$, the familiar Stirling number  of the second kind.In this note we first consider Stirling numbers of forests. We show that if $(F^{c(n)}_n)_{n\geq 0}$ is any sequence of forests with $F^{c(n)}_n$ having $n$ vertices and $c(n)=o(\sqrt{n/\log n})$ components, and if $X^{c(n)}_n$ is a random variable that takes value $k$ with probability proportional to $S(F^{c(n)}_n,k)$ (that is, $X^{c(n)}_n$ is the number of classes in a uniformly chosen partition of $F^{c(n)}_n$ into non-empty independent sets), then $X^{c(n)}_n$ is asymptotically normal, meaning that suitably normalized it tends in distribution to the standard normal. This generalizes a seminal result of Harper on the ordinary Stirling numbers. Along the way we give recurrences for calculating the generating functions of the sequences $(S(F^c_n,k))_{k \geq 0}$, show that these functions have all real zeroes, and exhibit three different interlacing patterns between the zeroes of pairs of consecutive generating functions.We next consider Stirling numbers of cycles. We establish asymptotic normality for the number of classes in a uniformly chosen partition of $C_n$ (the cycle on $n$ vertices) into non-empty independent sets. We give a recurrence for calculating the generating function of the sequence $(S(C_n,k))_{k \geq 0}$, and use this to give a direct proof of a log-concavity result that had previously only been arrived at in a very indirect way.

Cyclotomic Galois module structure and the second Chinburg invariant

1995 ◽  
Vol 117 (1) ◽  
pp. 57-82
Author(s):  
Victor P. Snaith
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Prime Power ◽  
Class Group ◽  
Number Fields ◽  
Affirmative Answer ◽  
Module Structure ◽  
Integral Group ◽  
Galois Module

AbstractWe study the second Chinburg invariant of a Galois extension of number fields. The Chinburg invariant lies in the class-group of the integral group-ring of the Galois group of the extension. A procedure is given whereby to evaluate the invariant in the case of the real cyclotomic case of regular prime power conductor and their subextensions of p-power degree. The invariant is shown to be zero in the latter cases, which yields new examples giving an affirmative answer to a question of Chinburg ([1], p. 358) which has come to be known as ‘Chinburg's Second Conjecture’ ([3], §4·2).

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