An upper bound for the real tensor rank and the real symmetric tensor rank in terms of the complex ranks

Abstract

Fix a real tensor of format . Here we prove that its real tensor rank is at most times its complex tensor rank. A similar bound is true for symmetric tensors and for any perfect field (e.g. a finite field).

Keywords:

• symmetric tensor rank
• tensor rank
• Segre variety
• Veronese variety
• real rank
• finite field

AMS Subject Classifications:

• 14N05
• 14Q05
• 15A69

Acknowledgement

The author was partially supported by MIUR and GNSAGA of INdAM (Italy).