  Given a positive integer $n$, a sufficient condition on a field is given for
bounding its Pythagoras number by $2^n+1$. The condition is satisfied for $n=1$
by function fields of curves over iterated formal power series fields over
$\mathbb{R}$, as well as by finite field extensions of
$\mathbb{R}(\!(t_0,t_1)\!)$. In both cases, one retrieves the upper bound $3$
on the Pythagoras number. The new method presented here might help to establish
more generally $2^n+1$ as an upper bound for the Pythagoras number of function
fields of curves over $\mathbb{R}(\!(t_1,\dots,t_n)\!)$ and for finite field
extensions of $\mathbb{R}(\!(t_0,\dots,t_n)\!)$.
