Factorization formulae on counting zeros of diagonal equations over finite fields

Cao, Wei; Sun, Qi

doi:10.1090/S0002-9939-06-08622-9

Factorization formulae on counting zeros of diagonal equations over finite fields
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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