A new family of exceptional polynomials in characteristic two

• 成果类型: Article
• 署名作者: Guralnick, Robert M.; Rosenberg, Joel; Zieve, Michael E.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• 发表日期: 2010
• 页码: 1361-1390
• 关键词: cyclic difference sets; permutation polynomials; hyperelliptic curves; power functions; conjecture

摘要

We produce a new family of polynomials f(X) over fields k of characteristic 2 which are exceptional, in the sense that f(X) - f(Y) has no absolutely irreducible factors in k[X, Y] except for scalar multiples of X - Y; when k is finite, this condition is equivalent to saying that the map alpha bar right arrow f(alpha) induces a bijection on an infinite algebraic extension of k. Our polynomials have degree 2(e-1)(2(e) - 1), where e > 1 is odd. We also prove that this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic.