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## Permanent Polya problem for additive surjective maps

Linear Algebra and its Applications. 2020. Vol. 599. P. 140-155.
Guterman A., Spiridonov I.A.

Let $M_{n}(\mathbb{F})$ denote the set of square matrices  of size $n$ over a field $\mathbb{F}$ with characteristics different from two. We say that the map  $f: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ is additive if $f(A+B) = f(A) + f(B)$ for all $A, B \in M_{n}(\mathbb{F})$. The main goal of this paper is to prove that for $n>2$ there are no additive surjective maps $T: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ such that $\per(T(A)) = \det(A)$ for all $A \in M_{n}(\mathbb{F})$. Also we show that an arbitrary additive surjective  map $T: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ which preserves permanent is linear and thus can be completely characterized.