Let $P_1(X,Y),\cdots,P_n(X,Y)$ be polynomials of $\mathbb C[X,Y]$ and $S$ be an infinite subset of $\mathbb C^2$ such that $P_1,\cdots,P_n$ vanish on $S$. My question: do there exist a polynomial $G\in\mathbb C[X,Y]$ and an infinite subset $T$ of $S$ such that for every integer $j$ ($1\le j\le n$) $P_j(X,Y)=G(X,Y)Q_j(X,Y)$ (with $Q_j(X,Y)\in\mathbb C[X,Y]$) and an index $i$ ($1\le i\le n$) with $Q_i$ not vanishing on $T$?

I tried to apply the Nullstellensatz, but I did not manage to prove this assertion.

Thanks in advance for any answer.

Let $P_1(X,Y),\cdots,P_n(X,Y)$ be polynomials of $\mathbb C[X,Y]$ not all zero and $S$ be an infinite subset of $\mathbb C^2$ such that $P_1,\cdots,P_n$ vanish on $S$. My question: do there exist a polynomial $G\in\mathbb C[X,Y]$ and an infinite subset $T$ of $S$ such that for every integer $j$ ($1\le j\le n$) $P_j(X,Y)=G(X,Y)Q_j(X,Y)$ (with $Q_j(X,Y)\in\mathbb C[X,Y]$) and an index $i$ ($1\le i\le n$) with $Q_i$ not vanishing on $T$?

I tried to apply the Nullstellensatz, but I did not manage to prove this assertion.

Thanks in advance for any answer.