On the solvability of systems of bilinear equations in finite fields
Author:
Le Anh Vinh
Journal:
Proc. Amer. Math. Soc. 137 (2009), 28892898
MSC (2000):
Primary 11L40, 11T30; Secondary 11E39
Published electronically:
May 4, 2009
MathSciNet review:
2506446
Fulltext PDF
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Additional Information
Abstract: Given sets and a nondegenerate bilinear form in , we consider the system of bilinear equations We show that the system is solvable for any , , given that the restricted sets are sufficiently large.
 1.
James
Arthur Cipra, Todd
Cochrane, and Christopher
Pinner, Heilbronn’s conjecture on Waring’s number (mod
𝑝), J. Number Theory 125 (2007), no. 2,
289–297. MR 2332590
(2008d:11116), http://dx.doi.org/10.1016/j.jnt.2006.12.001
 2.
D. Covert, D. Hart, A. Iosevich, and I. UriarteTuero, An analog of the FurstenbergKatznelsonWeiss theorem on triangles in sets of positive density in finite field geometries, preprint 2008, arXiv:0804.4894.
 3.
K.
Gyarmati and A.
Sárközy, Equations in finite fields with restricted
solution sets. II. (Algebraic equations), Acta Math. Hungar.
119 (2008), no. 3, 259–280. MR 2407038
(2009m:11207), http://dx.doi.org/10.1007/s1047400770350
 4.
D. Hart, Explorations of Geometric Combinatorics in Vector Spaces over Finite Fields, Ph.D. Thesis, Missouri University.
 5.
Derrick
Hart and Alex
Iosevich, Sums and products in finite fields: an integral geometric
viewpoint, Radon transforms, geometry, and wavelets, Contemp. Math.,
vol. 464, Amer. Math. Soc., Providence, RI, 2008,
pp. 129–135. MR 2440133
(2009m:11032), http://dx.doi.org/10.1090/conm/464/09080
 6.
Derrick
Hart and Alex
Iosevich, Ubiquity of simplices in subsets of vector spaces over
finite fields, Anal. Math. 34 (2008), no. 1,
29–38 (English, with English and Russian summaries). MR 2379694
(2008m:05296), http://dx.doi.org/10.1007/s104760080103z
 7.
D. Hart, A. Iosevich, D. Koh and M. Rudnev, Averages over hyperplanes, sumproduct theory in finite fields, and the ErdősFalconer distance conjecture, to appear in Trans. Amer. Math. Soc., arXiv:0707.3473.
 8.
D. Hart, A. Iosevich, D. Koh, S. Senger, and I. UriarteTuero, Distance graphs in vector spaces over finite fields, coloring, pseudorandomness and arithmetic progressions, preprint, 2008, arXiv:0804.3036.
 9.
M.
Krivelevich and B.
Sudakov, Pseudorandom graphs, More sets, graphs and numbers,
Bolyai Soc. Math. Stud., vol. 15, Springer, Berlin, 2006,
pp. 199–262. MR 2223394
(2007a:05130), http://dx.doi.org/10.1007/9783540324393_10
 10.
A.
Sárközy, On products and shifted products of residues
modulo 𝑝, Integers 8 (2008), no. 2, A9,
8. MR
2438294 (2009f:11034)
 11.
Igor
E. Shparlinski, On the solvability of bilinear equations in finite
fields, Glasg. Math. J. 50 (2008), no. 3,
523–529. MR 2451747
(2009j:11189), http://dx.doi.org/10.1017/S0017089508004382
 12.
L. A. Vinh, On a FurstenbergKatznelsonWeiss type theorem over finite fields, to appear in Ann. Comb., arXiv:0807.2849
 13.
L. A. Vinh, On kaleidoscopic pseudorandomness of finite Euclidean graphs, preprint, 2008, arXiv:0807.2689.
 14.
L. A. Vinh, On simplexes in dimensional vector spaces over finite fields, to appear in Proc. 21st FPSAC, 2009.
 15.
L. A. Vinh, Triangles in vector spaces over finite fields, to appear in Online J. Anal. Comb. (2009).
 16.
André
Weil, Numbers of solutions of equations in
finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508.
MR
0029393 (10,592e), http://dx.doi.org/10.1090/S000299041949092194
 1.
 J. A. Cipra, T. Cochrane and C. Pinner, Heilbronn's conjecture on Waring's number (mod ), J. Number Theory 125(2) (2007), 289297. MR 2332590 (2008d:11116)
 2.
 D. Covert, D. Hart, A. Iosevich, and I. UriarteTuero, An analog of the FurstenbergKatznelsonWeiss theorem on triangles in sets of positive density in finite field geometries, preprint 2008, arXiv:0804.4894.
 3.
 K. Gyarmati and A. Sárközy, Equations in finite fields with restricted solution sets, II (algebraic equations), Acta Math. Hungar. 119 (2008), 259280. MR 2407038
 4.
 D. Hart, Explorations of Geometric Combinatorics in Vector Spaces over Finite Fields, Ph.D. Thesis, Missouri University.
 5.
 D. Hart and A. Iosevich, Sums and products in finite fields: An integral geometric viewpoint, Contemp. Math. 464, Amer. Math. Soc., Providence, RI, 2008, pp. 129135. MR 2440133
 6.
 D. Hart and A. Iosevich, Ubiquity of simplices in subsets of vector spaces over finite fields, Anal. Math. 34(1) (2008), 2938. MR 2379694 (2008m:05296)
 7.
 D. Hart, A. Iosevich, D. Koh and M. Rudnev, Averages over hyperplanes, sumproduct theory in finite fields, and the ErdősFalconer distance conjecture, to appear in Trans. Amer. Math. Soc., arXiv:0707.3473.
 8.
 D. Hart, A. Iosevich, D. Koh, S. Senger, and I. UriarteTuero, Distance graphs in vector spaces over finite fields, coloring, pseudorandomness and arithmetic progressions, preprint, 2008, arXiv:0804.3036.
 9.
 M. Krivelevich and B. Sudakov, Pseudorandom graphs, in More Sets, Graphs and Numbers, Bolyai Soc. Math. Studies 15, Springer, Berlin, 2006, 199262. MR 2223394 (2007a:05130)
 10.
 A. Sárközy, On products and shifted products of residues modulo , Integers 8(2) (2008), A9. MR 2438294
 11.
 I. E. Shparlinski, On the solvability of bilinear equations in finite fields, Glasg. Math. J. 50 (2008), 523529. MR 2451747
 12.
 L. A. Vinh, On a FurstenbergKatznelsonWeiss type theorem over finite fields, to appear in Ann. Comb., arXiv:0807.2849
 13.
 L. A. Vinh, On kaleidoscopic pseudorandomness of finite Euclidean graphs, preprint, 2008, arXiv:0807.2689.
 14.
 L. A. Vinh, On simplexes in dimensional vector spaces over finite fields, to appear in Proc. 21st FPSAC, 2009.
 15.
 L. A. Vinh, Triangles in vector spaces over finite fields, to appear in Online J. Anal. Comb. (2009).
 16.
 A. Weil, Number of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497508. MR 0029393 (10:592e)
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Additional Information
Le Anh Vinh
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email:
vinh@math.harvard.edu
DOI:
http://dx.doi.org/10.1090/S000299390909947X
PII:
S 00029939(09)09947X
Keywords:
Bilinear equations,
finite fields
Received by editor(s):
December 1, 2008
Published electronically:
May 4, 2009
Communicated by:
Ken Ono
Article copyright:
© Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
