scholarly journals THE AUTOCORRELATION OF THE MOBIUS FUNCTION AND CHOWLA'S CONJECTURE FOR THE RATIONAL FUNCTION FIELD

2013 ◽  
Vol 65 (1) ◽  
pp. 53-61 ◽  
Author(s):  
D. Carmon ◽  
Z. Rudnick
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Möbius Function ◽  
Rational Function Field ◽  
Mobius Function

scholarly journals The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140311 ◽  
Author(s):  
Dan Carmon
Keyword(s):  
Finite Field ◽  
Rational Function ◽  
Function Field ◽  
Möbius Function ◽  
Rational Function Field ◽  
Mobius Function ◽  
Characteristic 2

We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field of characteristic 2, extending previous work in odd characteristic.

scholarly journals The differential Galois group of the rational function field

2021 ◽  
Vol 381 ◽  
pp. 107605
Author(s):  
Annette Bachmayr ◽  
David Harbater ◽  
Julia Hartmann ◽  
Michael Wibmer
Keyword(s):  
Galois Group ◽  
Rational Function ◽  
Function Field ◽  
Differential Galois Group ◽  
Rational Function Field

scholarly journals Mean value theorems for L-functions over prime polynomials for the rational function field

2013 ◽  
Vol 161 (4) ◽  
pp. 371-385 ◽  
Author(s):  
Julio C. Andrade ◽  
Jonathan P. Keating
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Mean Value ◽  
Rational Function Field ◽  
Mean Value Theorems

The Structure of Valuations of the Rational Function Field K(x)

1951 ◽  
Vol 71 (1) ◽  
pp. 102
Author(s):  
B. N. Moyls
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Rational Function Field

scholarly journals Indépendance algébrique de logarithmes en caractéristique P

2006 ◽  
Vol 74 (3) ◽  
pp. 461-470 ◽  
Author(s):  
Laurent Denis
Keyword(s):  
Zeta Function ◽  
Rational Function ◽  
Riemann Zeta Function ◽  
Function Field ◽  
Fundamental Period ◽  
Rational Function Field ◽  
Linearly Independent ◽  
The Riemann Zeta Function ◽  
Carlitz Module ◽  
Riemann Zeta

Let k be the rational function field over the field with q elements with characteristic p. Since the work of Carlitz we know in this situation the function ζ analog of the Riemann zeta function and the function Logφ analog of the usual logarithm. We will show two main results. Firstly, if ξ denotes the fundamental period of Carlitz module, we prove that ξ, ζ(1),…, ζ(p – 2) are algebraically independent over k. Secondly if α1,…, αn are rational elements (of degree less than q/(q − 1) to ensure convergence of the logarithm) such that Logφ α1,…, Logφ αn are linearly independent over k then they are algebraically independent over k. The point is to find suitable functions taking these values and for which Mahler's method can be used.

scholarly journals Square-free values of polynomials over the rational function field

2014 ◽  
Vol 135 ◽  
pp. 60-66 ◽  
Author(s):  
Zeév Rudnick
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Rational Function Field

scholarly journals Reduction theory for a rational function field

2003 ◽  
Vol 113 (2) ◽  
pp. 153-163 ◽  
Author(s):  
Amritanshu Prasad
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Reduction Theory ◽  
Rational Function Field

scholarly journals Zeta functions of a class of elliptic curves over a rational function field of characteristic two

1999 ◽  
Vol 68 (226) ◽  
pp. 823-834
Author(s):  
Ernst-Ulrich Gekeler ◽  
Rita Leitl ◽  
Bodo Wack
Keyword(s):  
Elliptic Curves ◽  
Rational Function ◽  
Function Field ◽  
Zeta Functions ◽  
Rational Function Field ◽  
Characteristic Two

scholarly journals The invariants of orthogonal group actions

1993 ◽  
Vol 48 (2) ◽  
pp. 313-319 ◽  
Author(s):  
Li Chiang ◽  
Yu-Ching Hung
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Rational Function ◽  
Linear Group ◽  
Function Field ◽  
Orthogonal Group ◽  
Group Actions ◽  
Rational Function Field ◽  
Transcendental Extension

Let Fq be the finite field of order q, an odd number, Q a non-degenerate quadratic form on , O(n, Q) the orthogonal group defined by Q. Regard O(n, Q) as a linear group of Fq -automorphisms acting linearly on the rational function field Fq(x1, …, xn). We shall prove that the invariant subfield Fq(x1,…, xn)O(n, Q) is a purely transcendental extension over Fq for n = 5 by giving a set of generators for it.

Defining transcendentals in function fields

2002 ◽  
Vol 67 (3) ◽  
pp. 947-956 ◽  
Author(s):  
Jochen Koenigsmann
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Algebraic Closure ◽  
Rational Function Field ◽  
First Order

AbstractGiven any field K, there is a function field F/K in one variable containing definable transcendental over K, i.e., elements in F / K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t).For the proof, diophantine ∅-definability ofK in F is established for any function field F/K in one variable, provided K is large, or K× /(K×)n is finite for some integer n > 1 coprime to char K.

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