If $p,q:\mathbb R^3\rightarrow\mathbb R$ are two polynomials, such that $\{p=0\}\cap\{q=0\}$ is twodimensional, does it follow that $p$ and $q$ have a common factor? (I believe it does.) How to prove that?
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Here is my answer. We consider the rimg $A=R[x,y,z]$ , $dimA=3$. We have $V((p))∩V((q))=V((p,q))$. If $gcd(p,q)=1$, then $ht(p,q)=2$. So $V((p))∩V((q))))=V((p,q))$ is not 2dimension. 

