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I know how to prove that for a commutative ring $R$, the units of the polynomial ring $R[x]$ are the polynomials $$p(x)=a_{0}+a_{1}x+\dots +a_{n}x^{n}$$ such that $a_{0}$ is a unit in $R$ and the remaining coefficients $a_{1},\dots ,a_{n}$ are nilpotent, i.e., satisfy $a_{i}^{N}=0$ for some $N$.

What can be said about the units of the polynomial ring in multiple variables, $R[x_1,\dots,x_n]$? Is there a similar characterization?

I was thinking of using that $R[x_1,\dots,x_n]= R[x_1,\dots,x_{n-1}] [x_n]$ and trying to apply induction, but I’m not sure how to proceed.

dahemar
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    There is no good characterization in the noncommutative case. See this: https://math.stackexchange.com/questions/147848/invertible-polynomials-over-non-commutative-rings – Mark Sep 25 '22 at 12:23
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    Well, you could say least initially consider $R[x,y]=R[x][y]$. What are the units there? – Arthur Sep 25 '22 at 12:30

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Your idea was good: using this question, you may easily prove by induction on $n\in\mathbb N$ that the units in $R[X_1,\dots,X_n]$ are the polynomials $$P(X)=\sum a_{k_1,\dots,k_n}X_1^{k_1}\dots X_n^{k_n}$$ such that (in $R$) $a_{0,\dots,0}$ is a unit and all other $a_{k_1,\dots,k_n}$'s are nilpotent.

See also Theorem 3.1 and 3.2 in NILPOTENTS, UNITS, AND ZERO DIVISORS FOR POLYNOMIALS by Keith Conrad

Anne Bauval
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