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

Drmota, Michael; Gutenbrunner, Georg

doi:10.5802/jtnb.481

The joint distribution of

Q

-additive functions on polynomials over finite fields

Drmota, Michael ¹ ; Gutenbrunner, Georg 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, Tome 17 (2005) no. 1, pp. 125-150.

Résumé
Abstract

Soient $K$ un corps fini et $Q \in K [T]$ un polynôme de degré au moins égal à 1. Une fonction $f$ sur $K [T]$ est dite (complètement) $Q$ -additive si $f (A + B Q) = f (A) + f (B)$ pour tous $A, B \in K [T]$ tels que $deg (A) < deg (Q)$ . Nous montrons que les vecteurs $(f_{1} (A), ..., f_{d} (A))$ sont asymptotiquement équirépartis dans l’ensemble image ${(f_{1} (A), ..., f_{d} (A)) : A \in K [T]}$ si les $Q_{j}$ sont premiers entre eux deux à deux et si les $f_{j} : K [T] \to K [T]$ sont $Q_{j}$ -additives. En outre, nous établissons que les vecteurs $(g_{1} (A), g_{2} (A))$ sont asymptotiquement indépendants et gaussiens si $g_{1}, g_{2} : K [T] \to ℝ$ sont $Q_{1}$ - resp. $Q_{2}$ -additives.

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 $f (A + B Q) = f (A) + f (B)$ , where $A, B \in K [T]$ and $deg (A) < deg (Q)$ . We prove that the values $(f_{1} (A), ..., f_{d} (A))$ are asymptotically equidistributed on the (finite) image set ${(f_{1} (A), ...,$ $f_{d} (A)) : A \in K [T]}$ if $Q_{j}$ are pairwise coprime and $f_{j} : K [T] \to K [T]$ are $Q_{j}$ -additive. Furthermore, it is shown that $(g_{1} (A), g_{2} (A))$ are asymptotically independent and Gaussian if $g_{1}, g_{2} : K [T] \to ℝ$ are $Q_{1}$ - resp. $Q_{2}$ -additive.

MR Zbl

DOI : 10.5802/jtnb.481

Affiliations des auteurs :

Drmota, Michael ¹ ; Gutenbrunner, Georg ¹

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

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     title = {The joint distribution of $Q$-additive functions on polynomials over finite fields},
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     pages = {125--150},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
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Drmota, Michael; Gutenbrunner, Georg. The joint distribution of $Q$-additive functions on polynomials over finite fields. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 125-150. doi : 10.5802/jtnb.481. https://www.numdam.org/articles/10.5802/jtnb.481/

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