Summary: In [Proc. Am. Math. Soc. 138, No. 5, 1559–1567 (2010; Zbl 1189.14065)], M. Marshall settled the strip conjecture, according to which every polynomial in \(\mathbb {R}[x,y]\), nonnegative on the strip \([-1,1]\times \mathbb {R}\), is a sum of squares and of squares times \(1-x^2\). We consider affine nonsingular curves \(C\) over \(\mathbb {R}\) with \(C(\mathbb {R})\) compact, and study the question of whether every \(f\) in \(\mathbb {R}[C][y]\), nonnegative on \(C(\mathbb {R})\times \mathbb {R}\), is a sum of squares in \(\mathbb {R}[C][y]\). We give an affirmative answer under the condition that \(f\) has only finitely many zeros in \(C(\mathbb {R})\times \mathbb {R}\). For \(C\) the circle \(x_1^2+x_2^2=1\), we prove the result unconditionally.
For the entire collection see [Zbl 1375.00094].