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Number of solutions of systems of homogeneous polynomial equations over finite fields


Authors: Mrinmoy Datta and Sudhir R. Ghorpade
Journal: Proc. Amer. Math. Soc. 145 (2017), 525-541
MSC (2010): Primary 14G15, 11T06, 11G25, 14G05; Secondary 51E20, 05B25
DOI: https://doi.org/10.1090/proc/13239
Published electronically: October 27, 2016
MathSciNet review: 3577858
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Abstract: We consider the problem of determining the maximum number of common zeros in a projective space over a finite field for a system of linearly independent multivariate homogeneous polynomials defined over that field. There is an elaborate conjecture of Tsfasman and Boguslavsky that predicts the maximum value when the homogeneous polynomials have the same degree that is not too large in comparison to the size of the finite field. We show that this conjecture holds in the affirmative if the number of polynomials does not exceed the total number of variables. This extends the results of Serre (1991) and Boguslavsky (1997) for the case of one and two polynomials, respectively. Moreover, it complements our recent result that the conjecture is false, in general, if the number of polynomials exceeds the total number of variables.


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Additional Information

Mrinmoy Datta
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Address at time of publication: Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK 2800, Kgs. Lyngby, Denmark
Email: mrinmoy.dat@gmail.com

Sudhir R. Ghorpade
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Email: srg@math.iitb.ac.in

DOI: https://doi.org/10.1090/proc/13239
Received by editor(s): July 17, 2015
Received by editor(s) in revised form: April 12, 2016
Published electronically: October 27, 2016
Additional Notes: The first author was partially supported by a doctoral fellowship from the National Board for Higher Mathematics, a division of the Department of Atomic Energy, Government of India.
The second author was partially supported by Indo-Russian project INT/RFBR/P-114 from the Department of Science & Technology, Government of India, and IRCC Award grant 12IRAWD009 from IIT Bombay.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society