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On the number of solutions of the linear equation in finite Carlitz modules

Author(s): Chih-Nung Hsu; Ting-Ting Nan
Journal: Proc. Amer. Math. Soc. 137 (2009), 2191-2200.
MSC (2000): Primary 11G09; Secondary 11T55, 11T24
Posted: January 13, 2009
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Abstract: We deduce an accurate formula for the number of solutions of the linear equation in generators of finite Carlitz modules, and the equation always has solutions except for some cases. Therefore, we have a criterion for the existence of the solutions of the linear equation. Moreover, we have a similar result in normal bases when we apply our main theorem to a special case.

References:

1.: L. Carlitz, Sums of Primitive Roots in a Finite Field, Duke Math. J., 19 (1952), 459-469. MR 0050628 (14:357c)
2.: L. Carlitz, Sets of Primitive Roots, Compositio Math., 13 (1956), 65-70. MR 0083002 (18:642e)
3.: D. Goss, Basic Structures of Function Field Arithmetic, Springer-Verlag (1996). MR 1423131 (97i:11062)
4.: H. W. Lenstra and R. J. Schoof, Primitive Normal Bases for Finite Fields, Math. Comp., 48 (1987), 217-231. MR 866111 (88c:11076)
5.: R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press (1997). MR 1429394 (97i:11115)

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Additional Information:

Chih-Nung Hsu
Affiliation: Department of Mathematics, National Taiwan Normal University, 88 Sec. 4, Ting-Chou Road, Taipei, Taiwan, Republic of China
Email: maco@math.ntnu.edu.tw

Ting-Ting Nan
Affiliation: Department of Mathematics, National Taiwan Normal University, 88 Sec. 4, Ting-Chou Road, Taipei, Taiwan, Republic of China
Email: ayanami-nan@math.ntnu.edu.tw

DOI: 10.1090/S0002-9939-09-09747-0
PII: S 0002-9939(09)09747-0
Received by editor(s): June 12, 2008,
Received by editor(s) in revised form: September 1, 2008
Posted: January 13, 2009
Communicated by: Ken Ono
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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