Consider $R_d=k[X_0,\ldots, X_n]_d$, the homogeneous polynomials of degree $d$ in $n+1$ variables. The dimension of $R$ viewed as $k$-vector space is $\binom{n+d}{d}$ so a general polynomial in $R$ has $\binom{n+d}{d}$ coefficients.
Now, let $d$ be divisible by $2$ and $3$ and $f\in R_{d/2}, g\in R_{d/3}$. Consider the polynomial $h = a_1\, f^2 +a_2\, g^3\in R_d$. Is it possible to compute the number of independent coefficients of $h$ for generic $a_i \in k$?
It has to be $\leq\binom{n+d}{d}$ but I do not figure out what the exact value is.