The joint distribution of Q -additive functions on polynomials over finite fields

Michael Drmota[1]; Georg Gutenbrunner[1]

  • [1] Inst. of Discrete Math. and Geometry TU Wien Wiedner Hauptstr. 8–10 A-1040 Wien, Austria

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 125-150
  • ISSN: 1246-7405

Abstract

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Let K be a finite field and Q ∈ K [ T ] a polynomial of positive degree. A function f on K [ T ] is called (completely) Q -additive if f ( A + B Q ) = f ( A ) + f ( B ) , where A , B ∈ K [ T ] and deg ( A ) < deg ( Q ) . We prove that the values ( f 1 ( A ) , ... , f d ( A ) ) are asymptotically equidistributed on the (finite) image set { ( f 1 ( A ) , ... , f d ( A ) ) : A ∈ K [ T ] } if Q j are pairwise coprime and f j : K [ T ] → K [ T ] are Q j -additive. Furthermore, it is shown that ( g 1 ( A ) , g 2 ( A ) ) are asymptotically independent and Gaussian if g 1 , g 2 : K [ T ] → ℝ are Q 1 - resp. Q 2 -additive.

How to cite

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Drmota, Michael, and Gutenbrunner, Georg. "The joint distribution of $Q$-additive functions on polynomials over finite fields." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 125-150. <http://eudml.org/doc/249423>.

@article{Drmota2005,
abstract = {Let $K$ be a finite field and $Q \in K[T]$ a polynomial of positive degree. A function $f$ on $K[T]$ is called (completely) $Q$-additive if $f(A +BQ) = f(A) + f(B)$, where $A, B \in K[T]$ and $\deg (A) &lt; \deg (Q)$. We prove that the values $(f_1(A), \ldots , f_d(A))$ are asymptotically equidistributed on the (finite) image set $\lbrace (f_1(A), \ldots ,$$ f_d(A)) : A \in K[T]\rbrace $ if $Q_j$ are pairwise coprime and $f_j:K[T]\rightarrow K[T]$ are $Q_j$-additive. Furthermore, it is shown that $(g_1(A), g_2(A))$ are asymptotically independent and Gaussian if $g_1,g_2:K[T]\rightarrow \mathbb\{R\}$ are $Q_1$- resp. $Q_2$-additive.},
affiliation = {Inst. of Discrete Math. and Geometry TU Wien Wiedner Hauptstr. 8–10 A-1040 Wien, Austria; Inst. of Discrete Math. and Geometry TU Wien Wiedner Hauptstr. 8–10 A-1040 Wien, Austria},
author = {Drmota, Michael, Gutenbrunner, Georg},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {-additive functions; polynomials over finite fields; distribution of -additive functions; joint distribution; convergence to the Gaussian normal distribution; correlation functions},
language = {eng},
number = {1},
pages = {125-150},
publisher = {Université Bordeaux 1},
title = {The joint distribution of $Q$-additive functions on polynomials over finite fields},
url = {http://eudml.org/doc/249423},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Drmota, Michael
AU - Gutenbrunner, Georg
TI - The joint distribution of $Q$-additive functions on polynomials over finite fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 125
EP - 150
AB - Let $K$ be a finite field and $Q \in K[T]$ a polynomial of positive degree. A function $f$ on $K[T]$ is called (completely) $Q$-additive if $f(A +BQ) = f(A) + f(B)$, where $A, B \in K[T]$ and $\deg (A) &lt; \deg (Q)$. We prove that the values $(f_1(A), \ldots , f_d(A))$ are asymptotically equidistributed on the (finite) image set $\lbrace (f_1(A), \ldots ,$$ f_d(A)) : A \in K[T]\rbrace $ if $Q_j$ are pairwise coprime and $f_j:K[T]\rightarrow K[T]$ are $Q_j$-additive. Furthermore, it is shown that $(g_1(A), g_2(A))$ are asymptotically independent and Gaussian if $g_1,g_2:K[T]\rightarrow \mathbb{R}$ are $Q_1$- resp. $Q_2$-additive.
LA - eng
KW - -additive functions; polynomials over finite fields; distribution of -additive functions; joint distribution; convergence to the Gaussian normal distribution; correlation functions
UR - http://eudml.org/doc/249423
ER -

References

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