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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Monodromy groups of rational functions - Michael Z
ieve (University of Michigan)
DTSTART;TZID=Europe/London:20220628T111500
DTEND;TZID=Europe/London:20220628T121500
UID:TALK175631AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/175631
DESCRIPTION:For each field $K$ of characteristic $0$\, we dete
rmine all degree-$n$ rational functions $f(X) \\in
K(X)$ for which the Galois group of the Galois cl
osure of $K(x)/K(f(x))$ is not $A_n$ or $S_n$.&nbs
p\; For many applications\, this means determining
all rational functions over $K$ which behave diff
erently from a typical rational function of the sa
me degree. \; We give applications to value di
stribution of meromorphic functions\, near-injecti
vity of rational functions over number fields\, an
d bijectivity of rational functions on subgroups o
f 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 pr
imitive groups.\nAbstract (in regular text): For e
ach field K of characteristic 0\, we determine all
degree-n rational functions f(X) in K(X) for whic
h the Galois group of the Galois closure of K(x)/K
(f(x)) is not A_n or S_n. \; For many applicat
ions\, this means determining all rational functio
ns over K which behave differently from a typical
rational function of the same degree. \; We gi
ve applications to value distribution of meromorph
ic functions\, near-injectivity of rational functi
ons over number fields\, and bijectivity of ration
al functions on subgroups of the multiplicative gr
oup of a finite field. \; The proofs rely on v
arious results classifying primitive groups with a
dditional properties\, and the topics lead to new
types of questions about primitive groups.
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:
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