What is known about rational points on the ideal of relations / syzygy ideal?

Question

What is known about rational points on the ideal of relations / syzygy ideal? Let $G$ be a finite group, with $|G|=n$. Then $G$ acts on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation (it permutes the $x_i$). Let $\mathbb{Q}[x_1,\cdots,x_n]^G = \mathbb{Q}[f_1,\cdots,f_m]$ be the invariant ring which is known to be finitely generated. Then in general the $f_i$ might fullfill some relations (when $G=S_n$ they do not). Those relations are captured in the ideal of relations: $I = \{ h \in \mathbb{Q}[y_1,\cdots,y_m] | h(f_1,\cdots,f_m) = 0 $ in $ \mathbb{Q}[x_1,\cdots,x_n]\}$ What is known about the ideal of relations? Do there alway exists rational points $(u_1,\cdots,u_m)\in \mathbb{Q}^m$ such that $h(u_1,\cdots,u_m) = 0$ for every $h\in I$?

Answer 1

The answer is yes, and it is very simple to see:

Just choose arbitrary $a_1,\cdots,a_n \in \mathbb{Q}$ and set $u_i := f_i(a_1,\cdots,a_n)$ for all $i=1,\cdots,m$. Then for $h \in I$ we have:

$h(u_1,\cdots,u_n) = h(f_1(a_1,\cdots,a_n),\cdots,f_m(a_1,\cdots,a_n)) = 0$

by definition of $h$.