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Number of Zeros of Diagonal Polynomials over Finite Fields
- Source :
- Finite Fields and Their Applications. 7:197-204
- Publication Year :
- 2001
- Publisher :
- Elsevier BV, 2001.
-
Abstract
- Let F be a finite field with q=pf elements, where p is a prime. Let N be the number of solutions (x1,…,xn) of the equation c1xd11+···+cnxdnn=c over the finite fields, where d1∣q−1, ciϵF*(i=1, 2,…,n), and cϵF. In this paper, we prove that if b1 is the least integer such that b1≥∑ni=1 (f/ri) (Di, p−1)/(p−1), then q[b1/f]−1∣N, where ri is the least integer such that di∣pri−1, Didi=pri−1, the (Di, p−1) denotes the greatest common divisor of Di and p−1, [b1/f] denotes the integer part of b1/f. If di=d, then this result is an improvement of the theorem that pb∣N, where b is an integer less than n/d, obtained by J. Ax (1969, Amer. J. Math.86, 255–261) and D. Wan (1988, Proc. AMS103, 1049–1052), under a certain natural restriction on d and n.
- Subjects :
- Discrete mathematics
Algebra and Number Theory
cyclotomic fields
Applied Mathematics
Gauss sum
Diagonal
General Engineering
diagonal polynomials over finite fields
Prime (order theory)
Theoretical Computer Science
Combinatorics
symbols.namesake
Finite field
Integer
Greatest common divisor
symbols
Engineering(all)
Mathematics
Subjects
Details
- ISSN :
- 10715797
- Volume :
- 7
- Database :
- OpenAIRE
- Journal :
- Finite Fields and Their Applications
- Accession number :
- edsair.doi.dedup.....38ac1e20d45f6631fca6a4fd3be438bd