Let $V$ be a $\mathbb{C}$-vector space, and let $f\_1,\dots,f\_n \in S^d(V^\*)$ be homogeneous polynomials of degree $d$ for which $V(f\_1,\dots, f\_n)=\{0\}$.


Must there exist a positive integer $k\geq d$ such that for all $v \in S^k(V^\*)$ there exists $g\_1,\dots, g\_n \in S^{k-d}(V^\*)$ for which $v=\sum\_{i=1}^n g\_i f\_i$?


I am inclined to guess that this is true, based on the projective Nullstellensatz, which tells us that for any finite set of linear forms $L=\{l\_1,\dots, l\_m\}$ that spans $V^\*$, there exists a positive integer $k\geq d$ such that for all $l\in L$ there exists $g\_1,\dots, g\_n \in S^{k-d}(V^\*)$ for which $l^k=\sum\_{i=1}^n g\_i f\_i$.



A variation on the projective Nullstellensatz
Since $V(f\_1,\dots,f\_n)=0$, by the Nullestellensatz
$$ \sqrt{ (f\_1,\dots, f\_n)} = I ( V( f\_1,\dots, f\_n)) = I(0) = (x\_1,\dots, x\_r)$$
where $x\_1,\dots, x\_r$ are a basis for $V^\*$.


So for each $i$ from $1$ to $r$, $x\_i^{e\_i} \in (f\_1,\dots, f\_n)$ for some $e\_i \in \mathbb N$.


It follows that $$(x\_1,\dots,x\_r)^{\sum\_{i=1}^n e\_i} \subseteq (f\_1,\dots, f\_n) $$ as each monomial in $(x\_1,\dots,x\_r)^{\sum\_{i=1}^r e\_i}$ is a multiple of $x\_i^{e\_i}$ for some $i$. (In fact, we can subtract $r-1$ from the exponent if desired).


So taking $k =\sum\_{i=1}^r e\_i$, for $n \geq k$, and $v$ homogeneous of degree $n$, we have $v\in (x\_1,\dots,x\_r)^n \subseteq (x\_1,\dots,x\_r)^k$, so $v = \sum\_{i=1}^n g\_i v\_i$, and we can take the degree $n-k$ part of each $g\_i$.


