Semi-definite representations for sets of cubics on the 2-sphere

Roland Hildebrand

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arXiv · 2103.13270

Semi-definite representations for sets of cubics on the 2-sphere

Roland Hildebrand

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Abstract

The compact set of homogeneous quadratic polynomials in $n$ real variables with modulus bounded by 1 on the unit sphere $S^{n-1}$ is trivially semi-definite representable. The compact set of homogeneous ternary quartics with modulus bounded by 1 on the unit sphere $S^2$ is also semi-definite representable. This suggests that the compact set of homogeneous ternary cubics with modulus bounded by 1 on $S^2$ 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 $S^2$.

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Roland Hildebrand. 2021-03-24. Semi-definite representations for sets of cubics on the 2-sphere. https://arxiv.org/abs/2103.13270

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