A class of retracts of polynomial algebras (Q2403981)

Let \(k\) be a field of characteristic zero and let \(k^{[n]}=k[x_1,\ldots,x_n]\) be the polynomial algebra in \(n\) variables over \(k.\) A subalgebra \(R\) of a \(k\)-algebra \(S\) is called retract of \(S\) if it is the image of a retraction(that is an idempotent endomorphism) of \(S.\) The authors are introducing the notion of an endomorphism of \(k^{[n]}\) with sparse homogeneous parts and show that any retraction of \(k^{[n]}\) with sparse homogeneous parts is conjugate to a canonical retraction. As a consequence, if \(\varphi\) is a retraction of \(k^{[n]}\) of the form \(\varphi=\varphi^{(1)}+\varphi^{(a)}+\ldots+\varphi^{(2a-2)}, a\geq 2\) where \(\varphi^{(d)}\) is the homogeneous part of degree \(d\) of \(\varphi\), then \(\varphi\) is conjugate to a canonical retraction. In particular, \(\varphi(k^{[n]})\) is isomorphic to a polynomial algebra.