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On the Number of Solutions of Polynomial Systems

✍ Scribed by M. Boguslavsky

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
247 KB
Volume
3
Category
Article
ISSN
1071-5797

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✦ Synopsis

We consider systems of homogenous polynomial equations of degree d in a projective space ‫ސ‬ m over a finite field ‫ކ‬ q . We attempt to determine the maximum possible number of solutions of such systems. The complete answer for the case r ϭ 2, d Ͻ q Ϫ 1 is given, as well as new conjectures about the general case. We also prove a bound on the number of points of an algebraic set of given codimension and degree. We also discuss an application of our results to coding theory, namely to the problem of computing generalized Hamming weights for q-ary projective Reed-Muller codes.

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