Abstract
We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the maximum value of the real rank is at most twice the smallest typical rank, which is equal to the (complex) generic rank.
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Acknowledgments
We thank E. Ballico, J. Buczyński, and A. de Paris for several very helpful comments. We also thank the referee for several helpful comments, including suggesting the reference [36], and A. Boralevi for providing us a copy of [36]. G. Blekherman was partially supported by Alfred P. Sloan Research Fellowship and NSF CAREER award DMS-1352073.
