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Let $ q \in \mathbb{Z}[x_1,\dots,x_n] $ be a polynomial. Is there a good way to determine if $$ q=p^2 $$ for some polynomial $ p $? The first thing that comes to mind is that if $ q=p^2 $ then $$ \frac{\partial q}{\partial x_i}=2p \frac{\partial p}{\partial x_i} $$ So $ p $ should be a factor of $ GCD(\frac{\partial q}{\partial x_1}, \dots \frac{\partial q}{\partial x_n}) $.

I don't know if this helps but in the example I'm working with $ q $ is a homogeneous degree $ 4 $ polynomial in $ 6 $ variables with integer coefficients (all the integers are pretty small, $ \leq 22 $).

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The coefficients of $x_i^4$ in $p$ will be the squares of the coefficients of $x_i^2$ in $q$. Because of signs, there are $2^6=64$ very similar possibilities.

Then use the coefficients of $x_i^3x_j$ in $p$ and $x_i^2$ in $q$ to find the coefficient of $x_ix_j$ in $q$.

If your answers are consistent, check that $q^2=p$

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  • $\begingroup$ Ok this easily shows that my polynomial is not square, thank you $\endgroup$ Commented Mar 2, 2023 at 15:19

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