Nonlinear determinant preserving maps on matrix algebras (Q2332408)

In 1897, \textit{G. Frobenius} [Berl. Ber. 1897, 994--1015 (1897; JFM 28.0130.01)] described the bijective linear maps \(\varphi \colon M_n(\mathbb{C}) \to M_n(\mathbb{C})\) on the \(n\times n\) matrices over the complex field that preserve the determinant. They either have the form \(\varphi (x) = uxv \) or \(\varphi (x)=ux^{t}v\), where \(u\) and \(v\) are matrices with \(\det (uv)=1\). \textit{G. Dolinar} and \textit{P. Šemrl} [Linear Algebra Appl. 348, No. 1--3, 189--192 (2002; Zbl 0998.15011)] gave an analogous description, without the assumption of linearity, for surjective maps satisfying \(\det(\varphi (x)+\lambda \varphi (y))=\det(x+\lambda y)\) for all matrices \(x,y\) and \(\lambda \in \mathbb{C}\). After a detailed introduction about previous results, the following is proved. Let \(\mathbb{F}\) be a field with at least \(n^2+1\) distinct elements and let \(\varphi, \psi \colon M_n(\mathbb{F}) \to M_n(\mathbb{F})\) be maps, at least one of them surjective, satisfying that \(\det(\varphi(x)+\psi (y)) = \det (x+y)\), for all \(x,y \in M_n(\mathbb{F})\). Then there exist \(x_0,u,v\in M_{n}(\mathbb{F})\), with \(\det(uv)=1\), such that either \[ \varphi (x)=u(x+x_0)y, \quad \psi (x) = u(x-x_0)y, \qquad (x,y,\in M_n (\mathbb{F})), \] or \[ \varphi (x)=u(x+x_0)^t y, \quad \psi (x) = u(x-x_0)^t y, \qquad (x,y,\in M_n (\mathbb{F})). \]