On the solvability of systems of bilinear equations in finite fields

Author:
Le Anh Vinh

Journal:
Proc. Amer. Math. Soc. **137** (2009), 2889-2898

MSC (2000):
Primary 11L40, 11T30; Secondary 11E39

Published electronically:
May 4, 2009

MathSciNet review:
2506446

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given sets and a non-degenerate bilinear form in , we consider the system of bilinear equations

**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**, 10.1016/j.jnt.2006.12.001**2.**D. Covert, D. Hart, A. Iosevich, and I. Uriarte-Tuero,*An analog of the Furstenberg-Katznelson-Weiss 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**, 10.1007/s10474-007-7035-0**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**, 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**, 10.1007/s10476-008-0103-z**7.**D. Hart, A. Iosevich, D. Koh and M. Rudnev,*Averages over hyperplanes, sum-product theory in finite fields, and the Erdős-Falconer distance conjecture*, to appear in Trans. Amer. Math. Soc., arXiv:0707.3473.**8.**D. Hart, A. Iosevich, D. Koh, S. Senger, and I. Uriarte-Tuero,*Distance graphs in vector spaces over finite fields, coloring, pseudo-randomness and arithmetic progressions*, preprint, 2008, arXiv:0804.3036.**9.**M. Krivelevich and B. Sudakov,*Pseudo-random graphs*, More sets, graphs and numbers, Bolyai Soc. Math. Stud., vol. 15, Springer, Berlin, 2006, pp. 199–262. MR**2223394**, 10.1007/978-3-540-32439-3_10**10.**A. Sárközy,*On products and shifted products of residues modulo 𝑝*, Integers**8**(2008), no. 2, A9, 8. MR**2438294****11.**Igor E. Shparlinski,*On the solvability of bilinear equations in finite fields*, Glasg. Math. J.**50**(2008), no. 3, 523–529. MR**2451747**, 10.1017/S0017089508004382**12.**L. A. Vinh,*On a Furstenberg-Katznelson-Weiss type theorem over finite fields*, to appear in Ann. Comb., arXiv:0807.2849**13.**L. A. Vinh,*On kaleidoscopic pseudo-randomness 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**, 10.1090/S0002-9904-1949-09219-4

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11L40,
11T30,
11E39

Retrieve articles in all journals with MSC (2000): 11L40, 11T30, 11E39

Additional Information

**Le Anh Vinh**

Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Email:
vinh@math.harvard.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-09947-X

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.