行列式强度的下界
Lower bounds on the strength of the determinant
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中文总结 AI 辅助
本文通过建立行列式强度和分区秩的下界，证明了行列式的强度与Birch秩的关系，并展示了分区秩的下界为n-o(n)。
中文摘要 AI 辅助
我们建立了行列式强度和分区秩的新的下界。对于每一个素数p，我们证明了精确恒等式[str(det_p)=p]。一个弱单调性论证，结合连续素数之间间隙的界，然后给出[str(det_n)≥(1-o(1))n^{0.475}]对于足够大的n。由于行列式det_n的Birch秩总是4，这给出了第一个显式家族，表明在Birch秩的背景下，强度的界对度数的依赖是不可避免的。将det_n视为其列的n-线性形式，我们还证明了其分区秩至少是不超过n的最大素数。因此，[n-n^{0.525}≤prk(det_n)≤n]对于所有足够大的n成立，因此行列式的分区秩是n-o(n)。证明引入了一种交理论方法来下界强度：一个短强度分解产生了一个在行列式超曲面补集上的处处非零截面，而PGL_n的Chow环中的非零上 Chern类则阻碍了这样的截面。
英文摘要
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 $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of $\mathrm{det}_n$ 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 $\mathrm{det}_n$ 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.
