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Counting polynomials with distinct zeros in finite fields.
- Source :
-
Journal of Number Theory . May2017, Vol. 174, p118-135. 18p. - Publication Year :
- 2017
-
Abstract
- Let F q be a finite field with q = p e elements, where p is a prime and e ≥ 1 is an integer. Let ℓ , n be two positive integers such that ℓ < n . Fix a monic polynomial u ( x ) = x n + u n − 1 x n − 1 + ⋯ + u ℓ + 1 x ℓ + 1 ∈ F q [ x ] of degree n and consider all degree n monic polynomials of the form f ( x ) = u ( x ) + v ℓ ( x ) , v ℓ ( x ) = a ℓ x ℓ + a ℓ − 1 x ℓ − 1 + ⋯ + a 1 x + a 0 ∈ F q [ x ] . For any non-negative integer k ≤ min { n , q } , let N k ( u ( x ) , ℓ ) denote the total number of v ℓ ( x ) such that u ( x ) + v ℓ ( x ) has exactly k distinct roots in F q , i.e. N k ( u ( x ) , ℓ ) = | { f ( x ) = u ( x ) + v l ( x ) | f ( x ) has exactly k distinct zeros in F q } | . In this paper, we obtain explicit combinatorial formulae for N k ( u ( x ) , ℓ ) when n − ℓ is small, namely when n − ℓ = 1 , 2 , 3 . As an application, we define two kinds of Wenger graphs called jumped Wenger graphs and obtain their explicit spectrum. [ABSTRACT FROM AUTHOR]
- Subjects :
- *POLYNOMIALS
*FINITE fields
*PRIME numbers
*INTEGERS
*GRAPH theory
Subjects
Details
- Language :
- English
- ISSN :
- 0022314X
- Volume :
- 174
- Database :
- Academic Search Index
- Journal :
- Journal of Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 120799724
- Full Text :
- https://doi.org/10.1016/j.jnt.2016.10.008