2026/07/23 by Qiyuan Chen, Yuhao Zhao
#math.AC #math.AG #math.CO
We establish new lower bounds for the strength and partition rank of the determinant. For every prime p, we prove the exact identity str(detp)=p. A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives str(detn)≥ (1-o(1))n0.475 for sufficiently large n. Since the Birch rank of detn 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 detn 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-n0.525≤ prk(detn)≤ 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 PGLn obstructs such a section.