Claims › ext:5f071e1221700a40 › line of work

Its line of work

The border determinantal complexity $\dcb(f)$ of a polynomial $f$ is the least $m$ such that $f$ is a limit of determinants of $m\times m$ matrices of affine-linear forms. We prove that for every $n\ge3$, over $\CC$, \[ \dcb\Big(\sum_{i=1}^n x_i^n\Big)\ \ge\ \frac{(n-1)^2}{4e}, \qquad \sdcb\Big(\sum_{i=1}^n x_i^n\Big)\ \ge\ \frac{(n-1)^2}{2e} \] in the ordinary and symmetric models respectively; both match the known $O(n^2)$ upper bounds up to the constant.

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