Nonnegative Polynomials and Moment Problems on Algebraic Curves
| Autor: | Baldi, Lorenzo, Blekherman, Grigoriy, Sinn, Rainer |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative polynomials $\pos$ on a smooth real projective curve $X$. We show that there is a duality between its faces and totally real effective divisors on $X$. This allows us to fully describe the face lattice in case $X$ has genus $1$. The dual cone $\pos^\vee$ is known as the moment cone, and it plays an important role in real analysis. We compute the Carath\'{e}odory number of $\pos^\vee$ for an elliptic normal curve $X$, which measures the complexity of quadrature rules of measures supported on $X$. This number can also be interpreted as a maximal typical Waring rank with nonnegative coefficients. Interestingly, the topology of the real locus of $X$ influences the Carath\'{e}odory number of $\pos^\vee$. We apply our results to truncated moment problems on affine cubic curves, where we deduce sharp bounds on the flat extension degree. Comment: 29 pages |
| Databáze: | arXiv |
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