Lower bounds on the strength of the determinant

Abstract

We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\mathrm{str}({\det }_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of ${\det }_n$ is always $4$, this gives the first explicit family showing that the dependence on the degree in bounds for strength in terms of Birch rank is unavoidable. Viewing ${\det }_n$ as an $n$-linear form in its columns, we also prove that its partition rank is at least the largest prime not exceeding $n$. Consequently, \[ n-n^{0.525}\le \operatorname{prk}(\mathrm{det}_n)\le n \] for all sufficiently large $n$, and hence the partition rank of the determinant is $n-o(n)$. The proof introduces an intersection-theoretic method for lower-bounding strength: a short strength decomposition produces a nowhere-vanishing section of a split vector bundle on the complement of the determinantal hypersurface, while a nonzero top Chern class in the Chow ring of ${\mathrm{PGL}}_n$ obstructs such a section.

arXiv comment16 pages. Comments are welcome!