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Monodromy groups of rational functions

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GRA2 - Groups, representations and applications: new perspectives

For each field $K$ of characteristic $0$, we determine all degree-$n$ rational functions $f(X) \in K(X)$ for which the Galois group of the Galois closure of $K(x)/K(f(x))$ is not $A_n$ or $S_n$.  For many applications, this means determining all rational functions over $K$ which behave differently from a typical rational function of the same degree.  We give applications to value distribution of meromorphic functions, near-injectivity of rational functions over number fields, and bijectivity of rational functions on subgroups of the multiplicative group of a finite field.  The proofs rely on various results classifying primitive groups with additional properties, and the topics lead to new types of questions about primitive groups. Abstract (in regular text): For each field K of characteristic 0, we determine all degree-n rational functions f(X) in K(X) for which the Galois group of the Galois closure of K(x)/K(f(x)) is not A_n or S_n.  For many applications, this means determining all rational functions over K which behave differently from a typical rational function of the same degree.  We give applications to value distribution of meromorphic functions, near-injectivity of rational functions over number fields, and bijectivity of rational functions on subgroups of the multiplicative group of a finite field.  The proofs rely on various results classifying primitive groups with additional properties, and the topics lead to new types of questions about primitive groups.

This talk is part of the Isaac Newton Institute Seminar Series series.

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