Semi-definite representations for sets of cubics on the 2-sphere
Résumé
The compact set of homogeneous quadratic polynomials in n real variables with modulus bounded by 1 on the unit sphere is trivially semi-definite representable. The compact set of homogeneous ternary quartics with modulus bounded by 1 on the unit sphere is also semi-definite representable. This suggests that the compact set of homogeneous ternary cubics with modulus bounded by 1 on the unit sphere is semi-definite representable. We deduce an explicit semi-definite representation of this norm ball. More generally, we provide a semi-definite description of the cone of inhomogeneous ternary cubics which are nonnegative on the unit sphere. This allows to incorporate nonnegativity conditions on polynomials in this space into semi-definite programs by transforming them into semi-definite constraints on the coefficient vector.