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Toward a salmon conjecture. (English) Zbl 1262.14056

The set of all tensors of type \(4\times 4\times 4\) of border rank \(4\) is the fourth secant variety \(\sigma _4(\mathbb {P}_{\mathbb {C}}^3\times \mathbb {P}_{\mathbb {C}}^3\times \mathbb {P}_{\mathbb {C}}^3)\) of the Segre variety \(\mathbb {P}_{\mathbb {C}}^3\times \mathbb {P}_{\mathbb {C}}^3\times \mathbb {P}_{\mathbb {C}}^3\subset \mathbb {P}_{\mathbb {C}}^{63}\). The set-theoretic salmon conjecture gives a conjectural set of equations cutting it as a set (E. Allman and J. Rhodes [Adv. in Appl. Math. 40, No. 2, 127–148 (2008; Zbl 1131.92046)]; J. M. Landsberg and L. Manivel [Commun. Algebra 36, No. 2, 405–422 (2008; Zbl 1137.14038)]). Here the authors prove with a high numerical degree of accuracy that this secant variety is cut out by 1728 equations of degree 5, 1000 equations of degree 6 and 8000 equations of degree 9. They conjecture that their equations generate the homogeneous ideal of \(\sigma _4(\mathbb {P}_{\mathbb {C}}^3\times \mathbb {P}_{\mathbb {C}}^3\times \mathbb {P}_{\mathbb {C}}^3)\). They discuss the meaning of their numerical tools and what is still missing to get a formal proof. They first need to prove that the set of all \(3 \times 3\times 4\) tensors of border rank 4 is cut out by polynomials of degree 6 and 9. After this paper the set-theoretic salmon conjecture was fully proved [S. Friedland and E. Gross, J. Algebra 356, No. 1, 374–379 (2012; Zbl 1258.14001)].

MSC:

14L30 Group actions on varieties or schemes (quotients)
13A50 Actions of groups on commutative rings; invariant theory
14M12 Determinantal varieties
20G05 Representation theory for linear algebraic groups
15A72 Vector and tensor algebra, theory of invariants
15A69 Multilinear algebra, tensor calculus
65H10 Numerical computation of solutions to systems of equations
68W30 Symbolic computation and algebraic computation
14Q99 Computational aspects in algebraic geometry
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