Classify the following polynomials as polynomials in one variable, two variables etc.

$x^{2}-2 x y+y^{2}+1$

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$x^{2}-2 x y+y^{2}+1$ is a polynomial in three variable.

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Classify the following polynomials as polynomials in one variable, two variables etc.

$x y+y z+z x$