Factorization formulae on counting zeros of diagonal equations over finite fields
HTML articles powered by AMS MathViewer
- by Wei Cao and Qi Sun PDF
- Proc. Amer. Math. Soc. 135 (2007), 1283-1291 Request permission
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
- James Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261. MR 160775, DOI 10.2307/2373163
- Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1625181
- L. K. Hua and H. S. Vandiver, Characters over certain types of rings with applications to the theory of equations in a finite field, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 94–99. MR 28895, DOI 10.1073/pnas.35.2.94
- Kenneth F. Ireland and Michael I. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, 1982. Revised edition of Elements of number theory. MR 661047, DOI 10.1007/978-1-4757-1779-2
- Jean-René Joly, Équations et variétés algébriques sur un corps fini, Enseign. Math. (2) 19 (1973), 1–117 (French). MR 327723
- Rudolf Lidl and Harald Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR 746963
- Charles Small, Diagonal equations over large finite fields, Canad. J. Math. 36 (1984), no. 2, 249–262. MR 749983, DOI 10.4153/CJM-1984-016-6
- Qi Sun and Da Qing Wan, On the solvability of the equation $\sum ^n_{i=1}x_i/d_i\equiv 0\;(\textrm {mod}\,1)$ and its application, Proc. Amer. Math. Soc. 100 (1987), no. 2, 220–224. MR 884454, DOI 10.1090/S0002-9939-1987-0884454-6
- Qi Sun and Da Qing Wan, On the Diophantine equation $\sum ^n_{i=1}x_i/d_i\equiv 0\pmod 1$, Proc. Amer. Math. Soc. 112 (1991), no. 1, 25–29. MR 1047008, DOI 10.1090/S0002-9939-1991-1047008-8
- Qi Sun, Da Qing Wan, and De Gang Ma, On the Diophantine equation $\sum ^n_{i=1}x_i/d_i\equiv 0$ ($\textrm {mod}\,1$) and its applications, Chinese Ann. Math. Ser. B 7 (1986), no. 2, 232–236. A Chinese summary appears in Chinese Ann. Math. Ser. A 7 (1986), no. 2, 240. MR 858601
- Sun Qi, On diagonal equations over finite fields, Finite Fields Appl. 3 (1997), no. 2, 175–179. MR 1444703, DOI 10.1006/ffta.1996.0173
- Zhi-Wei Sun, Exact $m$-covers and the linear form $\sum ^k_{s=1}x_s/n_s$, Acta Arith. 81 (1997), no. 2, 175–198. MR 1456240, DOI 10.4064/aa-81-2-175-198
- Da Qing Wan, Zeros of diagonal equations over finite fields, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1049–1052. MR 954981, DOI 10.1090/S0002-9939-1988-0954981-2
- André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508. MR 29393, DOI 10.1090/S0002-9904-1949-09219-4
- Jacques Wolfmann, New results on diagonal equations over finite fields from cyclic codes, Finite fields: theory, applications, and algorithms (Las Vegas, NV, 1993) Contemp. Math., vol. 168, Amer. Math. Soc., Providence, RI, 1994, pp. 387–395. MR 1291445, DOI 10.1090/conm/168/01716
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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1283-1291
- MSC (2000): Primary 11T24, 11T06; Secondary 11D72
- DOI: https://doi.org/10.1090/S0002-9939-06-08622-9
- MathSciNet review: 2276636