Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Point count divisibility for algebraic sets over $ {\mathbb{Z}}/p^\ell{\mathbb{Z}}$ and other finite principal rings


Author: Daniel J. Katz
Journal: Proc. Amer. Math. Soc. 137 (2009), 4065-4075
MSC (2000): Primary 11T06; Secondary 13M10
DOI: https://doi.org/10.1090/S0002-9939-09-10017-5
Published electronically: July 28, 2009
MathSciNet review: 2538567
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We determine the greatest common divisor of the cardinalities of the algebraic sets generated by collections of polynomials $ f_1,\ldots,f_t$ of specified degrees $ d_1,\ldots,d_t$ in $ n$ variables over a finite principal ring $ R$. This generalizes the theorems of Ax ($ t=1$, $ R$ a field), N. M. Katz ($ t$ arbitrary, $ R$ a field), and Marshall-Ramage ($ t=1$, $ R$ an arbitrary finite principal ring).


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

  • 1. Alan Adolphson and Steven Sperber, $ p$-adic estimates for exponential sums and the theorem of Chevalley-Warning, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 4, 545-556. MR 932797 (89d:11112)
  • 2. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39:4129)
  • 3. James Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255-261. MR 0160775 (28:3986)
  • 4. Wei Cao and Qi Sun, A reduction for counting the number of zeros of general diagonal equation over finite fields, Finite Fields Appl. 12 (2006), no. 4, 681-692. MR 2257089 (2007f:11033)
  • 5. -, Improvements upon the Chevalley-Warning-Ax-Katz-type estimates, J. Number Theory 122 (2007), no. 1, 135-141. MR 2287115
  • 6. C. Chevalley, Démonstration d'une hypothèse de M. Artin, Abh. Math. Sem. Univ. Hamburg 11 (1936), 73-75.
  • 7. Pedro L. del Angel R., A remark on the Hodge type of projective varieties of low degree, J. Reine Angew. Math. 449 (1994), 173-177. MR 1268584 (95b:14004)
  • 8. Jean-René Joly, Nombre de solutions de certaines équations diagonales sur un corps fini, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1549-A1552. MR 0282949 (44:183)
  • 9. Daniel J. Katz, On $ p$-adic estimates of weights in Abelian codes over Galois rings, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 2005.
  • 10. Nicholas M. Katz, On a theorem of Ax, Amer. J. Math. 93 (1971), 485-499. MR 0288099 (44:5297)
  • 11. Murray Marshall and Garry Ramage, Zeros of polynomials over finite principal ideal rings, Proc. Amer. Math. Soc. 49 (1975), 35-38. MR 0360541 (50:12989)
  • 12. Bernard R. McDonald, Finite rings with identity, Pure and Applied Mathematics, Vol. 28, Marcel Dekker Inc., New York, 1974. MR 0354768 (50:7245)
  • 13. O. Moreno and C. J. Moreno, An elementary proof of a partial improvement to the Ax-Katz theorem, Applied algebra, algebraic algorithms and error-correcting codes (San Juan, PR, 1993), Lecture Notes in Comput. Sci., vol. 673, Springer, Berlin, 1993, pp. 257-268. MR 1251983 (94k:11039)
  • 14. -, Improvements of the Chevalley-Warning and the Ax-Katz theorems, Amer. J. Math. 117 (1995), no. 1, 241-244. MR 1314464 (95j:11116)
  • 15. Oscar Moreno and Francis N. Castro, Divisibility properties for covering radius of certain cyclic codes, IEEE Trans. Inform. Theory 49 (2003), no. 12, 3299-3303. MR 2045808 (2005d:94215)
  • 16. -, Improvement on Ax-Katz's and Moreno-Moreno's theorems with coding theory applications, Proceedings of IEEE International Symposium on Information Theory, 2003, p. 132.
  • 17. -, Improvement of Ax-Katz's and Moreno-Moreno's results and applications, Int. J. Pure Appl. Math. 19 (2005), no. 2, 259-267. MR 2138044
  • 18. Oscar Moreno and Carlos J. Moreno, The MacWilliams-Sloane conjecture on the tightness of the Carlitz-Uchiyama bound and the weights of duals of BCH codes, IEEE Trans. Inform. Theory 40 (1994), no. 6, 1894-1907. MR 1322391 (96j:94029)
  • 19. Oscar Moreno, Kenneth W. Shum, Francis N. Castro, and P. Vijay Kumar, Tight bounds for Chevalley-Warning-Ax-Katz type estimates, with improved applications, Proc. London Math. Soc. (3) 88 (2004), no. 3, 545-564. MR 2044049 (2005g:11114)
  • 20. Bernard Morlaye, Équations diagonales non homogènes sur un corps fini, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1545-A1548. MR 0282948 (44:182)
  • 21. Marc Perret, On the number of points of some varieties over finite fields, Bull. London Math. Soc. 35 (2003), no. 3, 309-320. MR 1960941 (2003m:14036)
  • 22. Debin Ren, Qi Sun, and Pingzhi Yuan, Number of zeros of diagonal polynomials over finite fields, Finite Fields Appl. 7 (2001), no. 1, 197-204, dedicated to Professor Chao Ko on the occasion of his 90th birthday. MR 1803944 (2001k:11055)
  • 23. Stephen H. Schanuel, An extension of Chevalley's theorem to congruences modulo prime powers, J. Number Theory 6 (1974), 284-290. MR 0349637 (50:2130)
  • 24. Daqing Wan, Zeros of diagonal equations over finite fields, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1049-1052. MR 954981 (89i:11138)
  • 25. E. Warning, Bemerkung zur vorstehenden arbeit von Herrn Chevalley, Abh. Math. Sem. Univ. Hamburg 11 (1936), 76-83.
  • 26. Richard M. Wilson, An Ax-Katz-type theorem for congruences modulo powers of a prime, J. Number Theory (to appear).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11T06, 13M10

Retrieve articles in all journals with MSC (2000): 11T06, 13M10


Additional Information

Daniel J. Katz
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: katz.daniel.j@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-09-10017-5
Received by editor(s): July 10, 2007
Received by editor(s) in revised form: April 26, 2009
Published electronically: July 28, 2009
Additional Notes: This work is in the public domain
Communicated by: Ted Chinburg

American Mathematical Society