Automorphisms of the endomorphism semigroup of a polynomial algebra

What is it about?

We describe the automorphism group of the endomorphism semigroup $({\rm End}(K[x_1,\dots,x_n])$ of a ring $K[x_1,\dots,x_n]$ of polynomials over an arbitrary field $K$. A similar result is obtained for the automorphism group of the category of the finitely generated free commutative-associative algebras of the variety ${\scr{CA}}$ of commutative algebras. This solves two problems posed by B. I. Plotkin [Tr. Mat. Inst. Steklova 242 (2003), Mat. Logika i Algebra, 176–207; MR2054494 (Problems 12 and 15)]. "More precisely, we prove that if $\varphi \in{\rm Aut}\,{\rm End}(K[x_1,\dots,x_n])$ then there exists a semi-linear automorphism $s\colon K[x_1,\dots,x_n]\to K[x_1,\dots,x_n]$ such that $\varphi(g)=s\circ g\circ s^{-1}$ for any $g\in {\rm End}(K[x_1,\dots,x_n])$. This extends the result obtained by A. Berzins for an infinite field $K$.

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