Finiteness of Pythagoras numbers of finitely generated real algebras

In this paper, we establish infiniteness results and propose a conjecture regarding the Pythagoras number P(A) for finitely generated real algebras A. Let X↪Pn be an integral projective surface over R, with X~ its normalization, and s∈Γ(X,OX(1)) a nonzero section such that VX~(s)red is real reduced. We prove P(Γ(Xs))=∞. Consequently, the Pythagoras numbers of integral smooth affine curves over R are shown to be unbounded.