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) ).(c) 2023 Elsevier B.V. All rights reserved.