Abstract
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.
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Communicated by Guoyin Li.
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Hildebrand, R. Semi-definite Representations for Sets of Cubics on the Two-dimensional Sphere. J Optim Theory Appl 195, 666–675 (2022). https://doi.org/10.1007/s10957-022-02104-0
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DOI: https://doi.org/10.1007/s10957-022-02104-0