Say I have an anti-symmetric polynomial function $$f(x,y)=y^{m}-x^{m}$$
Where $m$ is some positive integer. Since it is anti-symmetric, we know that there is at least one way of writing it as a product of anti-symmetric factor $y-x$ with a symmetric function $S(x,y)$ $$f(x,y)=(y-x)S(x,y)$$ My question is, for a given $m$ in $y^{m}-x^{m}$, can we know what is highest power of $y-x$ that we can extract from it such that $S(x,y)$ doesn't have any factor of $y-x$ left in it.
$$y^m - x^m = (y-x)\left(y^{m-1} + y^{m-2}x + y^{m-3}x^2 + \ldots + yx^{m-2} + x^{m-1}\right)$$ Setting $y = x$ in the 2nd factor gives $mx^{m-1}$, which is not $0$ (unless we are working in a field where $m\equiv 0$). Therefore it cannot have $y - x$ as a factor.