Toward a salmon conjecture
Daniel J. Bates, Luke Oeding
TL;DR
The paper tackles the Allman salmon problem by combining a representation-theoretic construction of low-degree defining equations with a geometric framework for secant varieties, and corroborates the set-theoretic description via a Bertini-based numerical computation. Specifically, the authors identify degree-$6$ and degree-$9$ GL-invariant modules $M_{6}$ and $M_{9}$ whose common zero-set captures $\sigma_{4}(\mathbb{P}^{3}\times\mathbb{P}^{3}\times\mathbb{P}^{3})$ up to high numerical accuracy, building on Landsberg–Manivel’s approach corrected by Friedland and the inheritance principle. They show, in a concrete setting $A\cong B\cong \mathbb{C}^{3}$, $C\cong \mathbb{C}^{4}$, that $\sigma_{4}(\mathbb{P}A\times\mathbb{P}B\times\mathbb{P}C)$ is set-theoretically defined by $M_{6}$ and $M_{9}$, together with degree-$5$ equations inherited from a smaller case and subspace-variant equations, providing evidence toward the ideal-theoretic version of the result. The numerical computation with Bertini confirms the predicted decomposition into two components (the secant variety and a subspace variety) and demonstrates the feasibility and reliability of numerical algebraic geometry for proving statements in high-dimensional algebraic geometry, with implications for phylogenetic invariants and the salmon prize problem.
Abstract
By using a result from the numerical algebraic geometry package Bertini we show that (up to high numerical accuracy) a specific set of degree 6 and degree 9 polynomials cut out the secant variety $σ_{4}(\mathbb{P}^{2}\times \mathbb{P} ^{2} \times \mathbb{P} ^{3})$. This, combined with an argument provided by Landsberg and Manivel (whose proof was corrected by Friedland), implies set-theoretic defining equations in degrees 5, 6 and 9 for a much larger set of secant varieties, including $σ_{4}(\mathbb{P}^{3}\times \mathbb{P} ^{3} \times \mathbb{P} ^{3})$ which is of particular interest in light of the salmon prize offered by E. Allman for the ideal-theoretic defining equations.