Triangular derivations related to problems on affine \(n\)-space (Q2781288)

Let \(k\) be a field of characteristic zero. The authors study three famous problems of affine geometry:NEWLINENEWLINENEWLINE1. The embedding problem. Does for any polynomial embedding \(\alpha: k^r \to k^n,\;r<n,\;n\geq 2,\) there exist a \(k\)-automorphism \(\varphi\) of \(k^n\) such that \(\varphi\circ\alpha(u_1,... ,u_r) =( u_1,... ,u_r,0,... ,0)\)?NEWLINENEWLINENEWLINE2. The cancellation problem. Let \(V\) be an affine variety over \(k\) and \(V\times k\cong k^n,\;n\leq 2,\) as algebraic varieties. Does this imply \(V\cong k^{n-1}\)?NEWLINENEWLINENEWLINE3. The linearization problem. Does for any polynomial automorphism \(F: k^n \to k^n\) such that \(F^p=\text{id}_{k^n}\) (for some \(p\geq 1\)) there exist an automorphism \(\varphi\) of \(k^n\) such that \(\varphi^{-1}\circ F\circ\varphi\) is linear?NEWLINENEWLINENEWLINEThe authors describe known answers to these problems in many particular cases, relate these problems to each other and give equivalences of these problems to properties of the system of locally nilpotent derivations \(D=(D_1,... ,D_r)\) of the polynomial ring \(k[T,U_1,... ,U_r,X_1,... ,X_n]\) associated to a given polynomial map \(\alpha=(f_1,... ,f_n):k^n\to k^n,\) where NEWLINE\[NEWLINE D_i=f_{1U_i}\partial_{X_1}+... +f_{nU_i}\partial_{X_n}+T\partial_{U_i}. NEWLINE\]NEWLINE The authors describe also the Shastri example of an embedding NEWLINE\[NEWLINE\alpha:\mathbb C\to\mathbb C^3,\;\alpha(u)=(U^3-3U,U^4-4U^2,U^5-10U)NEWLINE\]NEWLINE as a potential counterexample to the embedding problem, and in consequence to the remaining problems.