Polynomials with PSL(2) monodromy

• 成果类型: Article
• 署名作者: Guralnick, Robert M.; Zieve, Michael E.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• 发表日期: 2010
• 页码: 1315-1359
• 关键词: exceptional polynomials; permutation polynomials; abhyankars conjecture; automorphism group; FIELDS; schur

摘要

Let k be a field of characteristic p > 0, let q be a power of p, and let u be transcendental over k. We determine all polynomials f is an element of k[X]\k[X-p] of degree q(q-1)/2 for which the Galois group of f(X)-u over k(u) has a transitive normal subgroup isomorphic to PSL2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f is an element of k[X] such that deg(f) is not a power of p and f is functionally indecomposable over k but f decomposes over an extension of k. Moreover, except for one ramification configuration (which is handled in a companion paper with Rosenberg), we describe all indecomposable polynomials f is an element of k[X] such that deg(f) is not a power of p and f is exceptional, in the sense that X - Y is the only absolutely irreducible factor of f(X)-f(Y) which lies in k[X, Y]. It is known that, when k is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of k.