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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!

1 Answers 1

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It's all about existence of common roots (which can easily be checked with Gröbner Basis).

If you want to check if your system has solution resolve the equations to get a set of polynomials and compute the Gröbner Basis (which then has to be $\neq 1$).

Check, if $P$ has at least one solution. If not, you're done.

else

Check if $Q$ has at least one solution. If not, you're done. (There is no solution AT ALL in q)

else

Check if $P \cup Q$ has at least one solution. If yes, you're done, because there is $a$ and $b$ where all polynomials of $P$ and $Q$ vanish.

else

you're done, because if there were $a,b$ for which all polynomials of $P$ and $Q$ are $=0$ this would also be true for all in $P \cup Q$ (obviously).

Mind that the last reasoning only holds because Q is independent of $x_1, ... x_m$ and P is independent of $y_1, ... y_n$