Abstract
Given a formula in the language of fields we use Galois stratification to establish an effective algorithm to estimate the number of points over finite fields that satisfy the formula
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[CDM] Z. Chatzidakis, L. van den Dries, and A. Macintyre,Definable sets over finite fields, to appear in Journal für die reine und angewandte Mathematik.
[D] L. van den Dries,A remark on Ax' theorem on solvability modulo primes, Mathematische Zeitschrift208 (1991), 65–70.
[FHJ] M.D. Fried, D. Haran and M. Jarden,Galois stratification over Frobenius fields, Advances of Mathematics51 (1984), 1–35.
[FJ] M.D. Fried and M. Jarden,Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete III11, Springer, Heidelberg, 1986.
[FS] M.D. Fried and G. Sacerdote,Solving diophantine problems over all residue class fields of a number field and all finite fields, Annals of Mathematics104 (1976), 203–233.
[HJ] D. Haran and M. Jarden,Bounded statements in the theory of algebraically closed fields with distinguished automorphisms, Journal für die reine und angewandte Mathematik337 (1982), 1–17.
[H] R. Hartshorne,Algebraic Geometry, Graduate Texts in Mathematics52, Springer, New York, 1977.
[J1] M. Jarden,Galois stratification for the elementary theory of finite prime fields, a manuscript, Tel Aviv, 1990.
[J2] M. Jarden,Algebraic dimension over Frobenius fields, to appear in Forum Mathematicum.
[L] S. Lang,Algebra, Addison-Wesley, Reading, 1970.
[Li] W. Litz,Die Anzahl der rationalen Punkte von Varietäten über einem endlichen Körper, Diplomarbeit, Heidelberg, 1975.
[LW] S. Lang and A. Weil,Number of points of varieties in finite fields, American Journal of Mathematics76 (1954), 819–827.
[M] H. Matsumura,Commutative Algebra, second edition, Benjamin/Cummings, Reading, Mass., 1980.
[R] M. Raynaud,Anneaux Locaux Henséliens, LNM169, Springer, New York, 1970.
[W] D. Wan,Hilbert sets and zeta function over finite fields, to appear in Journal für die reine und angewandte Mathematik.
[ZS] O. Zariski and P. Samuel,Commutative Algebra II, Springer, New York, 1975.
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This work was partially done while all three authors were members of the Institute for Advanced Studies in Jerusalem.
Parially supported by BSF grant #87-00038 and NSA grant MDA 904-91-H-0057.
Partially supported by grants from the German-Israeli Foundation for Scientific Research and Development.
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Fried, M.D., Haran, D. & Jarden, M. Effective counting of the points of definable sets over finite fields. Israel J. Math. 85, 103–133 (1994). https://doi.org/10.1007/BF02758639
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DOI: https://doi.org/10.1007/BF02758639