Toward a Salmon Conjecture

Abstract

Methods from numerical algebraic geometry are applied in combination with techniques from classical representation theory to show that the variety of 3×3×4 tensors of border rank 4 is cut out by polynomials of degree 6 and 9. Combined with results of Landsberg and Manivel, this furnishes a computational solution of an open problem in algebraic statistics, namely, the set-theoretic version of Allman’s salmon conjecture for 4×4×4 tensors of border rank 4. A proof without numerical computation was given recently by Friedland and Gross.

2000 AMS Subject Classification:

• 14L30
• 13A50
• 14M12
• 20G05
• 15A72
• 15A69
• 65H10
• 68W30
• 14Q99

Keywords:

• Salmon conjecture
• algebraic statistics
• representation theory
• numerical algebraic geometry
• Bertini

Acknowledgments

The authors would like to thank Shmuel Friedland, J. M. Landsberg, and Bernd Sturmfels for useful discussions throughout this project.

Daniel Bates was partially supported by NSF grant DMS–0914674. Luke Oeding was supported by National Science Foundation grant Award No. 0853000: International Research Fellowship Program (IRFP).