Let $F$ be a field, let $S$ be a subset of the polynomial ring $F[X_1,\dotsc,X_n]$, where each polynomial in $S$ has degree at most $d$, and let $I$ be the ideal generated by $S$ (the degree of a polynomial is the maximum of the degrees of its monomials).
I am looking for a small subset $T$ of $S$ that generates $I$.
Since there are ${d+n\choose n}$ monomials of degree $\leq d$, I know that $S$ is contained in an $F$-vector space of degree $\leq{d+n\choose n}$, and so taking $T$ to be a maximal linearly independent subset of $S$ works, with the cardinality guarantee $|T|\leq {d+n\choose n}$.
Can we guarantee an even smaller $T$? If it helps - think a fixed $n$ and $d\to\infty$.
Edit: A more precise question: For a fixed $n$, my reasoning about shows that we can guarantee $|T|\leq Cd^{n}$, $C$ constant. By Mees de Vries' comment, taking $S$ to be the set of monomials of degree $d$, we see that we cannot do better than $Cd^{n-1}$. What is the best exponent? $n$? $n-1$? Something in between?