Progress on Strassen's Conjecture

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Strassen's conjecture for symmetric tensors is advanced by proving additivity for forms that are linearly computable, using apolar methods: compute f⊥, choose an ideal generated by linear forms and evaluate the quotient f⊥:I+(l) to get a concrete lower bound. Monomial rank is computed by an explicit formula and coprime monomials in disjoint variables are additive (examples give ranks 12 and 4, sum 16 or 12+12=24 in variations). The approach uses the Landsberg–Teitler singularity bound, succeeds for many reducible and binary cases, and also identifies cubic examples where the linear method provably fails.

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1. 0:00Introduction
2. 0:20Matrix multiplication context
3. 1:04Tensors and elementary operations
4. 2:26Strassen's breakthrough
5. 4:04Significance of the result
6. 6:05The conjecture explained
7. 8:04Formalizing the conjecture
8. 9:54Truth and border rank
9. 11:24Symmetric tensor conjecture
10. 12:57Difficulties in computation
11. 13:53Lower bound approaches
12. 15:24Polarity theory basics
13. 17:31Landsberg-Titler technique
14. 19:37Algebraic approach
15. 22:12Monomial case study
16. 24:08Linear computability
17. 28:31Proving the conjecture
18. 32:50Examples of computable forms
19. 38:41Non-computable examples
20. 46:04Final discussion and wrap-up