Abstract
Artin solved Hilbert’s 17th problem by showing that every positive semidefinite polynomial can be realized as a sum of squares of rational functions. Pfister gave a bound on the number of squares of rational functions: if p is a positive semi-definite polynomial in n variables, then there is a polynomial q so that q 2 p is a sum of at most 2n squares. As shown by D’Angelo and Lebl, the analog of Pfister’s theorem fails in the case of Hermitian polynomials. Specifically, it was shown that the rank of any multiple of the polynomial \(\|Z\|^{2d} \equiv (\sum_j|z_j|^2)^d\) is bounded below by a quantity depending on d. Here we prove that a similar result holds in a free *-algebra.
Mathematics Subject Classification. 16S10, 16W10.
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References
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Dedicated to Professor Helton, on the occasion of his 65th birthday
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Harrison, M. (2012). Pfister’s Theorem Fails in the Free Case. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_14
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DOI: https://doi.org/10.1007/978-3-0348-0411-0_14
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