On the solutions of the equation in a finite field
Authors:
Wen Fong Ke and Hubert Kiechle
Journal:
Proc. Amer. Math. Soc. 123 (1995), 13311339
MSC:
Primary 11D79; Secondary 11T30
MathSciNet review:
1234628
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Abstract 
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Additional Information
Abstract: An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on m and the characteristic of the field, holds.
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 J. R. Clay, Circular block designs from planar near rings, Ann. Discrete Math. 37 (1988), 95106. MR 931309 (89g:05019)
 [2]
 , Nearrings: geneses and applications, Oxford Univ. Press, Oxford, 1992. MR 1206901 (94b:16001)
 [3]
 J. R. Clay and H. Kiechle, Linear codes from planar near rings and Möbius planes, Algebras Groups Geom. 10 (1993), 333344. MR 1250686 (94k:51006)
 [4]
 L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, Amer. J. Math. 57 (1935), 391424. MR 1507083
 [5]
 , Congruences involving only eth powers, Acta Arith. 1 (1936), 161167.
 [6]
 O. B. Faircloth, On the number of solutions of some general types of equations in a finite field, Canad. J. Math. 4 (1952), 343351. MR 0047694 (13:915d)
 [7]
 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), 9499. MR 0028895 (10:515c)
 [8]
 , On the number of solutions of some trinomial equations in a finite field, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 477481. MR 0032679 (11:329g)
 [9]
 K. F. Ireland and M. I. Rosen, A classical introduction to modern number theory, second ed., SpringerVerlag, Berlin, Heidelberg, and New York, 1990. MR 1070716 (92e:11001)
 [10]
 J.R. Joly, Équations et variétés algébriques sur un corps fini, Enseign. Math. (2) 19 (1973), 1117.
 [11]
 W.F. Ke, Structures of circular planar nearings, Ph.D. dissertaton, Univ. Arizona, Tucson, 1992.
 [12]
 W.F. Ke and H. Kiechle, Combinatorial properties of ring generated circular planar nearings (submitted).
 [13]
 D. H. Lehmer, The number of solutions of a certain congruence involving the sum of like powers, Utilitas Math. 39 (1991), 6589. MR 1119764 (92f:11005)
 [14]
 H. H. Mitchell, On the congruence in a Galois field, Ann. of Math. (2) 18 (1917), 120131. MR 1503593
 [15]
 M. C. Modisett, A characterization of the circularity of certain balanced incomplete block designs, Ph.D. dissertation, Univ. Arizona, Tucson, 1988.
 [16]
 , A characterizatin of the circularity of balanced incomplete block designs, Utilitas Math. 35 (1989), 8394. MR 992392 (90c:05028)
 [17]
 Q. Sun, The diagonal equation over a finite field , Sichuan Daxue Xuebao 26 (1989), 159162. (Chinese) MR 1017018 (90i:11145)
 [18]
 A. Weil, Number of solutions of equations in a finite field, Bull. Amer. Math. Soc. 55 (1949), 497508. MR 0029393 (10:592e)
 [19]
 J. Wolfmann, The number of solutions of certain diagonal equations over finite fields, J. Number Theory 42 (1992), 247257. MR 1189504 (94j:11055)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512346284
PII:
S 00029939(1995)12346284
Article copyright:
© Copyright 1995
American Mathematical Society
