Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Factorization formulae on counting zeros of diagonal equations over finite fields


Authors: Wei Cao and Qi Sun
Journal: Proc. Amer. Math. Soc. 135 (2007), 1283-1291
MSC (2000): Primary 11T24, 11T06; Secondary 11D72
Published electronically: November 14, 2006
MathSciNet review: 2276636
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ N$ be the number of solutions $ (u_1,\ldots,u_n)$ of the equation $ a_1u_1^{d_1}+\cdots+a_nu_n^{d_n}=0$ over the finite field $ F_q$, and let $ I$ be the number of solutions of the equation $ \sum_{i=1}^nx_i/d_i\equiv 0\pmod{1}, 1\leqslant x_i\leqslant d_i-1$. If $ I>0$, let $ L$ be the least integer represented by $ \sum_{i=1}^nx_i/d_i, 1\leqslant x_i\leqslant d_i-1$. $ I$ and $ L$ play important roles in estimating $ N$. Based on a partition of $ \{d_1,\dots,d_n\}$, we obtain the factorizations of $ I, L$ and $ N$, respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for $ N$ in some special cases.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11T24, 11T06, 11D72

Retrieve articles in all journals with MSC (2000): 11T24, 11T06, 11D72


Additional Information

Wei Cao
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China
Email: caowei433100@vip.sina.com

Qi Sun
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08622-9
PII: S 0002-9939(06)08622-9
Keywords: Jacobi sum, Gauss sum, diagonal equation, finite fields
Received by editor(s): July 19, 2005
Received by editor(s) in revised form: December 21, 2005
Published electronically: November 14, 2006
Additional Notes: This work was partially supported by the National Natural Science Foundation of China, Grant #10128103.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.