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Consider two polynomials $P_1$, $P_2$ in polynomial ring $K[\mathbf{x}]$ such that the (sum)-degrees of $P_1, P_2$, are both equal to $1.$ That is, in effect, they are both linear (or more precisely affine). Let $I$ be the ideal generated by $P_1, P_2$.

I am trying to figure out if a polynomial $P,$ which is also linear, can belong to the radical ideal $\sqrt{I}$. I see two methods to approaching this question:

1) I can use Gaussian elimination to understand if $P$ is a $K$-linear combination of $P_1, P_2$. However, does this lead to the right answer always? That is, if I determine that $P$ cannot be written as $a_1P_1 + a_2 P_2,$ where $a_1, a_2 \in K,$ then is it necessarily true that $P \notin \sqrt{I}$?

2) Based on Hilbert's Nullstellensatz, I can try to solve for $q_1 P_1 + q_2 P_2 = P^L$ for some $L.$ However, I don't know if there is some way of bounding the degrees of $q_1, q_2$ in this case. Furthermore, if the first method does apply, I don't see how this relates to the above equation.

Any insights or references will help. Thanks!

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Let's work geometrically. Asking whether $P\in \sqrt{I}$ is equivalent to asking whether $(P)\subset \sqrt{I}$ which is equivalent to asking if $V(P)\supset V(I)$. This is equivalent to asking whether $P$ vanishes on the linear subspace that is determined by the vanishing of $P_1,P_2$. This question is pure linear algebra, so Gaussian elimination will do the job.