Rapidly increasing power substitutions isolate one highest x_n-term

Write $f={\sum }_{\alpha =({\alpha }_1,\dots ,{\alpha }_n)}{c}_{\alpha }{x}_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}$ with only finitely many nonzero coefficients. Choose $N$ larger than every exponent ${\alpha }_i$ occurring with $c_{\alpha }\mathrm{\ne }0$.

For each exponent vector $\alpha$, define the weight $w(\alpha )={\alpha }_n+{\alpha }_{1}N+{\alpha }_2{N}^2+\cdots +{\alpha }_{n-1}{N}^{n-1}\mathrm{.}$ Because every ${\alpha }_i<N$, base-$N$ expansion is unique, so distinct exponent vectors have distinct weights.

After substituting $x_i+x_n^{N^i}$ for $x_i$, the monomial $x_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}$ contributes the term $c_{\alpha }{x}_n^{w(\alpha )}$ to the highest $x_n$-power coming from that monomial; all other terms from its binomial expansion have smaller $x_n$-power. Let ${\alpha }^{\ast }$ be the exponent vector with maximal weight among those with $c_{\alpha }\mathrm{\ne }0$. By step 2.1 this ${\alpha }^{\ast }$ is unique, so $c_{{\alpha }^{\ast }}{x}_n^{w({\alpha }^{\ast })}$ is the unique highest $x_n$-term of the transformed polynomial.

The transformed polynomial is therefore nonzero and has a unique highest $x_n$-term with nonzero coefficient. Multiplying by the inverse of that coefficient makes it monic in $x_n$.