I want to prove that for an integer $n\geq 2$, $f_{n}\left ( x,y \right )$ and $f_{n-1}\left ( x,y \right )$ are two homogenuous polynomials without common divisor whose degree are n and n-1 respectively, how to prove that $f_{n}\left ( x,y \right )+f_{n-1}\left ( x,y \right )$ is irreducible?
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The sum is not homogeneous, hence the hypothetical factors can't both be homogeneous. If only one of them is homogeneous, then it will be the common divisor, and if both are, then the degrees of the terms in the product won't be limited to $n$ and $n-1$. So it seems. – Ivan Neretin Sep 13 '21 at 10:10
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Could you please tell me why "If only one of them is homogeneous, then it will be the common divisor"? thank you – YuerCauchy Sep 13 '21 at 10:17
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Let's see. Say it has degree $k$. Then the terms of the other factor can only have the degrees $n-k$ and $n-k-1$, otherwise some extra degrees will pop up in the result. Now, the terms with degree $n-k-1$ will give $f_{n-1}$, and those with degree $n-k$ will give $f_n$, so our hypothetical homogeneous factor of the sum is in fact a factor of both terms. – Ivan Neretin Sep 13 '21 at 10:36