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 2020-01-18 17:04:03 +0100 commented answer Testing whether polynomial is in algebra of other polynomials Hey. Thanks a lot for your answer! I still need to play around with it some more. In the meantime, do you think there's any way to also spit out how, for example, $x_0x_1+x_0x_2+x_1x_2$ came from the given algebra, i.e. what combination of stuff in our algebra yields $x_0x_1+x_0x_2+x_1x_2$? 2020-01-18 15:04:44 +0100 received badge ● Nice Question (source) 2020-01-18 14:48:00 +0100 received badge ● Student (source) 2020-01-18 11:29:57 +0100 asked a question Testing whether polynomial is in algebra of other polynomials A collection $\Sigma$ of polynomials is an algebra if: (1) $\lambda f + \eta g \in \Sigma$ for any $f,g \in \Sigma, \lambda,\eta \in \mathbb{R}$ and (2) $f,g \in \Sigma$ implies $fg \in \Sigma$. We say that $P$ is in the algebra of ${P_1,\dots,P_n}$ if $P$ is in the smallest algebra containing $P_1,\dots,P_n$. I was wondering if there was a way to check whether a given $P$ as in the algebra of a given collection $P_1,\dots,P_n$. Example: take $n \ge 1$ and let $P_1 = x_1+\dots+x_n, P_2 = x_1^2+\dots+x_n^2,\dots P_n = x_1^n+\dots+x_n^n$. Then all $n$ of the following symmetric functions are in the algebra generated by $P_1,\dots,P_n$: $$x_1+\dots+x_n$$ $$x_1x_2+\dots+x_{n-1}x_n$$ $$x_1x_2x_3+\dots+x_{n-2}x_{n-1}x_n$$ $$\dots$$ $$x_1\dots x_n$$ I'd appreciate any help.