The problem of determining a set of generators of the ring of invariants of the group $\textrm{SL}_2$ acting on the complex $n+1$-dimensional vector space of binary $n$-ic forms is known to be very hard and is open for $n > 10$. However we *do* know thanks to Hilbert that this is a finitely generated ring over $\mathbb{C}$. When $n = 2$ or $3$, the ring of invariants is generated by the discriminant of the binary (quadratic/cubic) form and when $n = 4$, we have two generators $I$ and $J$, both $\mathbb{Z}$- polynomials in the coefficients of the binary quartic form of degrees $2$ and $3$ respectively. This begs the general question -- even if one cannot exhibit a set of generators for the complex ring of invariants, can one show that there exists such a set of integral/rational polynomials (equivalent by scaling) in the coefficients of the binary form? Intuitively, I would guess that the answer is *yes*, but I don't know if there's a way to see this nor could I find a reference for it.

Thanks!