A generalization of Ore’s theorem on polynomials
| Autor: | Alexander V. Anashkin |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Equioscillation theorem
Discrete mathematics Applied Mathematics Discrete orthogonal polynomials 010102 general mathematics 01 natural sciences Classical orthogonal polynomials 010104 statistics & probability Difference polynomials Macdonald polynomials Hahn polynomials Wilson polynomials Orthogonal polynomials Discrete Mathematics and Combinatorics 0101 mathematics Mathematics |
| Zdroj: | Discrete Mathematics and Applications. 26:255-258 |
| ISSN: | 1569-3929 0924-9265 |
| Popis: | Let GF(q) be the field of q elements and Vn (q) denote the n-dimensional vector space over the field GF(q). The linearized polynomial that corresponds to the polynomial f ( x ) = x n − ∑ i = 0 n − 1 c i x i $f(x) = {x^n} - \sum\limits_{i = 0}^{n - 1} {{c_i}{x^i}} \;$ over the field GF(q) is the polynomial F ( x ) = x q n − ∑ i = 0 n − 1 c i x q i $F(x) = {x^{{q^n}}} - \sum\limits_{i = 0}^{n - 1} {{c_i}{x^{{q^i}}}}$ . Let Tf denote the transformation of the vector space Vn (q) determined by the rule T f ( u 0 , . . . , u n − 2 , u n − 1 ) = ( u 1 , . . . , u n − 1 , ∑ i = 0 n − 1 c i u i ) ${T_f}\left( {({u_0},...,{u_{n - 2}},{u_{n - 1}})} \right) = ({u_1},...,{u_{n - 1}},\sum\limits_{i = 0}^{n - 1} {{c_i}{u_i}} )$ . It is shown that if c 0 ≠ 0, then the graph of the transformation Tf is isomorphic to the graph of the transformation Q: α → αq on the set of all roots of the polynomial F(x) in its splitting field. In this case the graph of the transformation Tf consists of cycles of lengths 1 ≤ d 1 ≤ d 2 ≤ ... ≤ dr if and only if the polynomial F(x) is the product of r + 1 irreducible factors of degrees 1, d 1, d 2, ... dr . |
| Databáze: | OpenAIRE |
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