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SUMMARY:Sums of squares in function fields of curves and the Lam-Pfister c
onjecture.
DTSTART:20241016T120000Z
DTEND:20241016T130000Z
DTSTAMP:20241102T021200Z
UID:indico-event-12707@indico.math.cnrs.fr
DESCRIPTION:Speakers: Gonzalo Manzano Flores (Universidad de Chile)\n\nThe
Pythagoras number of a field is defined as the smallest positive integer
n such that each sum of squares in the field is a sum of n squares. For ex
ample\, the Pythagoras number of the field of rational numbers Q is 4 (for
example 7 is a sum of 4 squares\, but NOT of 3 squares in Q)\, which was
proven by Euler in 1751\, while the Pythagoras number of the rational func
tion fields in one variable Q(X) is 5 (for example the polynomial X^2+7 is
a sum of 5 squares\, but NOT of 4 squares)\, which was shown by Y. Pourch
et. Another basic example: the Pythagoras number of the field of real numb
ers R is 1\, and it can easily be shown that the Pythagoras number of the
rational function fields in one variable R(X) is 2. Thus\, T.Y. Lam (and l
ater A. Pfister) surmised the following: for an arbitrary field K\, can we
bound the Pythagoras number of the field of rational functions in one var
iable K(X) in terms of the Pythagoras number of its base field K? In gener
al\, this question still remains open. In this talk\, I will show a soluti
on to this problem in the particular case where the base field is already
a field of rational functions in one variable with the additional property
that each sum of squares in such a field is a sum of two squares. I will
also do a historical overview of some famous results related to this invar
iant in the case of function fields\, for example\, "Hilbert's Problem No.
17". \n\nhttps://indico.math.cnrs.fr/event/12707/
LOCATION:Salle Pellos (207) ( Institut de Mathématiques de Toulouse\, bâ
t. 1R2)
URL:https://indico.math.cnrs.fr/event/12707/
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