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Let $K$ be commutative ring. Assume that for natural $n$ there are $n$ nilpotent elements $y_i \in K$ satisfying $y_i^2=0, y_i y_j=y_j y_i \ne 0$ and $\prod_1^ny_i \neq 0$.

Is it possible to compute in time $\exp o(n)$: $$ f=\prod_{i=1}^n\left(\sum_{j=1}^na_{i,j}y_j\right). $$ for $a_{i,j} \in K$?

Very likely the answer is negative because of @Aaron's answer.

Treating $y_i$ as variables $f$ is homogeneous and is a single monomial.

Example with many nilpotent elements, but inefficient computation is $\mathbb{Z}[x_1,\ldots x_n]/(x_1^2,x_2^2\ldots x_n^2)$

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  • $\begingroup$ You would also want $\prod_1^ny_i \neq 0.$ $\endgroup$
    – Aaron Meyerowitz
    Commented Apr 18, 2019 at 14:47
  • $\begingroup$ @AaronMeyerowitz Indeed, thanks. I edited. $\endgroup$
    – joro
    Commented Apr 18, 2019 at 17:02
  • $\begingroup$ How does Aaron's answer to the other question not settle this up to Valiant's perm vs. det conjecture? $\endgroup$
    – Joshua Grochow
    Commented Apr 19, 2019 at 2:13

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