Laurent normalization by a monomial change of variables ($X_j=Y_j{Y}_r^{N^j}(j<r)$)

A nonzero quotient of an $r$-variable Laurent polynomial ring is finite over an embedded Laurent polynomial ring in $m\le r$ variables. Given a nonzero Laurent relation, choose $N$ so that weights $(N,N^2,\dots ,N^{r-1},1)$ separate its finite exponent support. The displayed invertible monomial substitution makes its extreme $Y_r$ coefficients units in the remaining Laurent variables. Normalize the relation to a monic polynomial with unit constant term; both $Y_r$ and $Y_r^{-1}$ are integral over the image of the $(r-1)$-variable subring. Induction and transitivity of finite extensions prove the assertion, over finite fields as well as infinite fields. An infinite-dimensional quotient has $m\ge 1$.