报告题目: Geometric invariants of complex polynomials
报告人: 付伟博 (中国科大华罗庚班2012级)
时间：2015年9月23号（周三）下午2:30―4:00
地点：管研楼1318
摘要: Two of the most important geometric invariants of a complex polynomial are its monodromy group and its ramification type. In topology, a fundamental theorem by Rene Thom describes all possibilities for the ramification type of a complex polynomial. In Galois theory, a much-used theorem by Walter Feit and Peter Muller describes all possibilities for the monodromy group of an indecomposable complex polynomial. Here we say that a polynomial is indecomposable if it cannot be written as the functional composition of lower-degree polynomials; thus, indecomposable polynomials are the “prime” polynomials under composition, and it turns out that many questions about arbitrary polynomials can be resolved if they can be solved for indecomposable polynomials. We prove a result which brings together the results of Thom and Feit�Muller by determining all possibilities for the pair (monodromy group of f, ramification type of f) where f varies over all indecomposable complex polynomials. Our proof involves a delicate reduction to the case of polynomials with only two (finite) critical values, a reinterpretation and resolution of the latter case in terms of a question about bicolored trees, and methods from group theory and Galois theory. Beyond the intrinsic importance of determining all possibilities for the fundamental invariants of an indecomposable polynomial, our result provides a new tool which helps resolve many questions about polynomials arising in dynamical systems, complex analysis, number theory, algebraic geometry, and other areas.
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