Bounding the Pythagoras number of a field by 2n + 1

Given a positive integer n, a sufficient condition on a field is given for bounding its Pythagoras number by 2n+1. The condition is satisfied for n=1 by function fields of curves over iterated formal power series fields over R, as well as by finite field extensions of R((t0,t1)). In both cases, one retrieves the upper bound 3 on the Pythagoras number. The new method presented here might help to establish more generally 2n+1 as an upper bound for the Pythagoras number of function fields of curves over R((t1,…,tn)) and for finite field extensions of R((t0,…,tn)).