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SUMMARY:Monodromy groups of rational functions - Michael Zieve (University
  of Michigan)
DTSTART:20220628T101500Z
DTEND:20220628T111500Z
UID:TALK175631@talks.cam.ac.uk
DESCRIPTION:For each field $K$ of characteristic $0$\, we determine all de
 gree-$n$ 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 $S_n$.&nbsp\; For ma
 ny applications\, this means determining all rational functions over $K$ w
 hich behave differently from a typical rational function of the same degre
 e.&nbsp\; We give applications to value distribution of meromorphic functi
 ons\, near-injectivity of rational functions over number fields\, and bije
 ctivity of rational functions on subgroups of the multiplicative group of 
 a finite field.&nbsp\; The proofs rely on various results classifying prim
 itive groups with additional properties\, and the topics lead to new types
  of questions about primitive groups.\nAbstract (in regular text): For eac
 h field K of characteristic 0\, we determine all degree-n rational functio
 ns 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.&nbsp\; For many applications\, this means determ
 ining all rational functions over K which behave differently from a typica
 l rational function of the same degree.&nbsp\; We give applications to val
 ue distribution of meromorphic functions\, near-injectivity of rational fu
 nctions over number fields\, and bijectivity of rational functions on subg
 roups of the multiplicative group of a finite field.&nbsp\; The proofs rel
 y on various results classifying primitive groups with additional properti
 es\, and the topics lead to new types of questions about primitive groups.
LOCATION:Seminar Room 2\, Newton Institute
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