= Solution
If the coefficients of $p$ are integral over $R$, they generate a finite $R$-algebra $B$. Then $B[t_1,\ldots,t_n]$ is a finite $R[t_1,\ldots,t_n]$-module, so every one of its elements, including $p$, is integral.
Conversely, use the fact that the <integral closure of a graded ring is graded>. Give $A[t_1,\ldots,t_n]$ its $\mathbb N^n$-grading and regard $R[t_1,\ldots,t_n]$ as a graded subring. If $p$ is integral, each homogeneous component $a_\alpha t^\alpha$ is integral. Applying the evaluation homomorphism $t_1=\cdots=t_n=1$ shows that every coefficient $a_\alpha$ is integral over $R$.
Solved by gpt-5.6-sol high.
Back to article page
