Automorphisms of the endomorphism semigroup of a polynomial algebra

Abstract

We describe the automorphism group of the endomorphism semigroup End ( K [ x 1 , … , x n ] ) of ring K [ x 1 , … , x n ] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative–associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15]. More precisely, we prove that if φ ∈ Aut End ( K [ x 1 , … , x n ] ) then there exists a semi-linear automorphism s : K [ x 1 , … , x n ] → K [ x 1 , … , x n ] such that φ ( g ) = s ∘ g ∘ s − 1 for any g ∈ End ( K [ x 1 , … , x n ] ) . This extends the result obtained by A. Berzins for an infinite field K.