I'm currently experimenting with polynomial ideals and Gröbner bases, and I seem to be lacking some terminology/understanding.
I have two systems of polynomial equations $P$ and $Q$ over a field $\mathbb F$.
$P$ uses variables $\{a,b,x_1, ... x_m\}$
$Q$ uses variables $\{a,b,y_1, ... y_n\}$
I want to know if there are any values of $\{a,b\}$ for which at least one solution exists in $P$ but no solutions exist in $Q$.
It feels like there should be something "about" $\{a,b\}$ that I can extract from $P$ and $Q$ which I can then compare, but I'm not sure what that would be called!