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We suggest the following slight generalization of Keller's theorem: If $f:\\mathbb{C}[x,y]\\to \\mathbb{C}[x,y]$ is a morphism having an invertible Jacobian, and if there exist $n \\geq 1$, $a \\in \\mathbb{C}(f(x),f(y))^*$ and $b \\in \\mathbb{C}(f(x),f(y))$ such that $(ax +b)^n \\in \\mathbb{C}(f(x),f(y))$, then $f$ is invertible. 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If we add the assumption that $\\mathbb{C}(f(x),f(y))=\\mathbb{C}(x,y)$, then $f$ is invertible; this result is due to O. H. Keller (1939). We suggest the following slight generalization of Keller's theorem: If $f:\\mathbb{C}[x,y]\\to \\mathbb{C}[x,y]$ is a morphism having an invertible Jacobian, and if there exist $n \\geq 1$, $a \\in \\mathbb{C}(f(x),f(y))^*$ and $b \\in \\mathbb{C}(f(x),f(y))$ such that $(ax +b)^n \\in \\mathbb{C}(f(x),f(y))$, then $f$ is invertible. 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