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Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of smaller degrees?

Are there any simple criteria to identify cases when such $m$ does not exist?

Is it possible that more than one suitable values of $m$ exist?

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The polynomial $(x_1+\cdots+x_n)^2+m$ becomes factorisable whenever $m=-k^2$ for some integer $k$. –  Aakumadula Mar 3 '13 at 6:39
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About your first question - there need not be any simple criteria to say when such $m$ does not exist. Take e.g polynomial $x^2 + ny + m$. As long as $n \neq 0$, no value of $m$ will make this factorisable. Examples of this kind suggests that a simple criteria in terms of coefficients of the polynomial may not exist. –  Amit Mar 3 '13 at 7:32
    
For $n\geq 2$, the polynomial $(x_1^2+\cdots+x_n^2)+m$ is never reducible. –  Name Mar 3 '13 at 7:33
    
@Aakumadula: Thanks for answering the third question! –  Max Alekseyev Mar 3 '13 at 8:32
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@Amit, on the contrary, such examples all follow from Newton polytope considerations. If the Newton polytope is not decomposable as a Minkowski sum, then such an $m$ does not exist. –  Gjergji Zaimi Mar 3 '13 at 13:46
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