Nonnegative Quadratics over a Quadrant with a Bilinear Constraint

We study quadratic polynomials that are nonnegative on the non-compact set \[ F:=\{(x_1,x_2)\in\mathbb R^2:\ x_1\ge 0,\ x_2\ge 0,\ x_1x_2\le 1\}. \] All extreme rays of the cone of nonnegative quadratic polynomials are characterized on this set. The characterization allows us to study parameterized valid inequalities for quadratic convexifications involving $F$, which yields a semidefinite representation of its lifted convex hull in the quadratic space. Our analysis on extreme ray characterization separates the positive-semidefinite (PSD) and non-PSD branches, reduces the latter to boundary nonnegativity, and classifies the boundary contacts of the relevant extreme rays. By reparameterization, we also find a non-trivial and non-permutation-symmetric six-dimensional linear section of the cone of nonnegative homogeneous ternary octics that are sum-of-squares. We also show that our lifted convex hull result yields a degree-bounded preordering certificate of nonnegative quadratics on $F$ and a degree-bounded certificate for a family of nonnegative quartics on the half-strip.