Sums of squares on curves and surfaces

We study sums of higher even powers in the coordinate rings of singular planar curves $x^M=y^m$ for coprime positive integers $m<M$. We then show that the $2s$-Pythagoras number of real algebras of the form $\mathbb{R}[x,y,\sqrt{f_1},\sqrt{f_2},\dots ,\sqrt{f_n}]$ are infinite, under some mild assumptions on the polynomials $f_1,f_2,\dots ,f_n\in \mathbb{R}[x,y]$. We prove that all of the higher even Pythagoras numbers are finite for the ring of $0$-regulous functions on a $0$-regulous variety. We then show that the codimension of the bad set of order $2n$, for $n>1$, can be of codimension $2$, contrary to the quadratic case.