Séminaire de Géométrie Algébrique et Singularités
# Sums of squares in function fields of curves and the Lam-Pfister conjecture.

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Salle Pellos (207) ( Institut de Mathématiques de Toulouse, bât. 1R2)
### Salle Pellos (207)

#### Institut de Mathématiques de Toulouse, bât. 1R2

Description

The 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 example, 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 function 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. Pourchet. Another basic example: the Pythagoras number of the field of real numbers 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 later A. Pfister) surmised the following: for an arbitrary field K, can we bound the Pythagoras number of the field of rational functions in one variable K(X) in terms of the Pythagoras number of its base field K? In general, this question still remains open. In this talk, I will show a solution 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 invariant in the case of function fields, for example, "Hilbert's Problem No. 17".