Let $P\_1,\ldots, P\_d, Q\_1, \ldots, Q\_k \in \mathbb{C}[x\_0,\ldots, x\_n]$ be homogenous polynomials of degree at most $r$.


Assume that $P\_1 \cdot P\_2 \cdots P\_{d-1} \cdot P\_d \in \langle Q\_1, \ldots, Q\_k \rangle$. 
Here $\langle h\_1, \ldots, h\_s \rangle$ is the ideal with the generators $h\_1, \ldots, h\_s$. 


Is it true that for some $\{ i\_1, \ldots, i\_f \} \subseteq \{1, \ldots, d\}$ the polynomial $P\_{i\_1} \cdots P\_{i\_f} \in \langle Q\_1, \ldots, Q\_k \rangle$, where $f= f(k,r)$?



Generators of an ideal with small degree
Yes. The point is that many invariants of the ideal $I=(Q\_1,\dots,Q\_k)$ can be bounded depending only on $k$ and $r$. It is pure luck that the following paper has collected many of them in a very convenient Proposition 4.6. 
<http://www-personal.umich.edu/~asnowden/papers/genstillman-071517.pdf>


In particular, you can find a primary decomposition of $I = I\_1\cap\dots \cap I\_l$ such that each of the $I\_i$ is $\mathfrak p\_i$-primary and the number of generators of $I\_i$ as well as degrees and $l$ itself are bounded by some function of $k$ and $r$ (using part 6 of the Prop. cited above). 


So the problem reduces to the case when $I$ is $\mathfrak p$-primary. But there is a $B$ such that $\mathfrak p^B\subseteq I$, and this number can also be bounded on the degrees and number of generators of $I$ (part 10 of loc. cit.). Such $B$ works. 


