5 problems

For $n>m$, let $X_{11}^{n-m}\mathrm{per}_m$ be the padded permanent, viewed as an element of $\mathsf{Sym}^n\mathbb{C}^{n^2}$, and let ${{\mathcal{D}}et}_n$ be the Euclidean (equiv…

For a polynomial $f$, let $\mathrm{dc}(f)$ denote the least size of a matrix of affine linear forms whose determinant equals $f$. Let $\mathrm{per}_m$ denote the permanent of an…

For $n>m$, let $Z_{n,m}$ be the orbit closure of the padded permanent $X_{11}^{n-m}\mathrm{per}_m$ in $\mathsf{Sym}^n\mathbb{C}^{n^2}$, and let ${{\mathcal{D}}et}_n$ be the orbit c…

Let $\mathrm{dc}(\mathrm{per}_m)$ be the determinantal complexity of the $m\times m$ permanent. For integers $d,n,m$, let $\gamma_{\lambda,d,n,m}$ and $\delta_{\lambda,d,n}$ denote…

Constructed-family determinantal lower-bound conjecture. If $a>0$ is small enough, then, with high probability, $f_n$ cannot be expressed as a symbolic determinant of size at most…