How to verify that $P \in \langle Q_1, \dots, Q_m \rangle$ in $\mathbb C[x_1, \dots , x_n]$?

Question

For $P, Q_1, \dots, Q_m \in \mathbb C[x_1, \dots, x_n]$, how to verify that $$P \in \langle Q_1, \dots, Q_m \rangle?$$

What are known algorithms to write $P$ as a combination of the $Q_1, \dots, Q_m$?

What if we replace $\mathbb C$ with another field?

My question comes from this one.

Answer 1

There is Buchberger's algorithm for finding a Gröbner basis.