Theatre Faculty Publications
Document Type
Article
Publication Date
Spring 5-6-2026
Abstract
We present a computational certificate for three families of explicit polyno- mial equations whose zero locus contains the fourth secant variety σ4(P3×P3×P3) inside P63. The three families are: 192 degree-5 Strassen commutation equa- tions (M5); 160 degree-6 equations lifted from the Bates–Oeding generators of σ4(P2 × P2 × P3) via the Landsberg–Manivel–Friedland theorem (M6); and 64 degree-9 Ottaviani 9 × 9 determinant equations (M9). All 416 generators are explicit polynomials in the coordinate ring Z[Zijk | i, j, k ∈ {0, 1, 2, 3}]. We verify by exact integer arithmetic that every generator vanishes on rank-4 test tensors (six independent deterministic test points) and is nonzero on rank-5 and rank-6 tensors. A Macaulay2 finite-field computation over Z/32003 inde- pendently confirms vanishing at a rank-4 point and shows that the Jacobian matrixofJ=(M5 |M6 |M9)atthatpointhasrank24,equaltotheex- pected codimension of σ4(P3×P3×P3). The set-theoretic version of the salmon conjecture was resolved by Friedland and Friedland–Gross; the contribution here is a fully explicit, reproducible generator set and verification pipeline directed at the still-open ideal-theoretic question. Full scheme-theoretic equality J = I(σ4(P3×P3×P3)) is not claimed; the remaining certificate steps (Hilbert series, saturation, associated primes, Gr ̈obner basis) require HPC resources beyond a standard workstation. All construction code is publicly reproducible.
Recommended Citation
Trevino, John, "Toward the Salmon Prize: A Computational Certificate for the M5, M6, and M9 Equation Families of σ4(P3 × P3 × P3)" (2026). Theatre Faculty Publications. 21.
https://scholarworks.utrgv.edu/the_fac/21
