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Let $f_1,\ldots,f_m \in \mathbb{F}[x_1,\ldots,x_n]$ and suppose $\hat{f}_i = f_i$ $\bmod \langle x_1,\ldots,x_n\rangle^3$ (i.e. the linear and quadratic part of $f_i$). Then if $\hat{f}_1,\ldots,\hat{f}_n$ are algebraically independent would that imply $f_1,\ldots,f_n$ are algebraically independent?

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$x^2 + y$ and $x^4 + 2x^2 y + y^2$ are algebraically dependent, but $x^2 + y$ and $y^2$ are not.

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    $\begingroup$ Thank you, I didn't expect such a surprisingly simple counter example. $\endgroup$ Commented Oct 13, 2023 at 9:44
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    $\begingroup$ First one I tried ;) $\endgroup$ Commented Oct 13, 2023 at 12:47

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