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Y u r i B u r d a U n i v e r s i t y o f T o r o n t o
y b u r d a @ m a t h . t o r o n t o . e d u
Topological Methods in Klein’s Resolvent Problem
A M S S e s s i o n o n D i f f e r e n t i a l A l g e b r a i c G e o m e t r y a n d G a l o i s T h e o r y
J a n u a r y 6 , 2 0 1 2
Klein’s resolvent problem
n=3 k=1
n=4 k=2
n=5 k≤2 k≥2
Hermite (1861) Kronecker, Klein (1888)
n=6 k≤3 Joubert (1876) n k ≥[n/2] Buhler, Reichstein,
Serre (1997)
n=7 k=4 Duncan (2010)
We’ll see: k ≥[n/2]
C o v e r i n g s o v e r T o r i a n d
T o p o l o g i c a l A p p r o a c h t o K l e i n ' s R e s o l v e n t P r o b l e m h t t p : / / a r x i v . o r g / a b s / 1 1 0 7 . 3 4 4 4
( s u b m i t t e d t o “ T r a n s f o r m a t i o n G r o u p s ” )
Y u r i B u r d a y b u r d a @ m a t h . t o r o n t o . e d u
Questions?
Additional results
� Can add any algebraic functions with monodromy group of odd order
� Can allow taking branches � Estimates for arbitrary algebraic functions � Generic function of k parameters of degree at least
2k can’t be simplified