I'm working in the following book "Computational invariant theory" second edition by Harm Derksen and Gregor Kemper. The book contains the following theorem on page 72
Let $R$ be a finitely generated algebra over a Noetherian commutative ring, and let $G$ be a finite group acting by automorphisms on $R$ fixing $K$ elementwise. Then $R^G$ is finitely generated as a $K-$algebra.
Note that $R^G$ denotes the elements of $R$ with are invariant under the action. The proof starts like this
Let $x_1,...,x_r$ be generators of $R$. The polynomial $$ \prod_{\sigma \in G} (T- \sigma \cdot x_i) = T^m + a_{i,1}T^{m-1} + \dots + a_{i,m-1}T + a_{i,m} \in R^G[T]$$ provides an integral equation for $x_i$ over $R^G$.
I searched through the book to find a definition of "an integral equation over $R^G$" but I can't find it. So I was wondering if someone could clarify what the meaning is.