Lemma 10.115.2. Let R be a ring. Let g \in R[x_1, \ldots , x_ n] be an element which is nonconstant, i.e., g \not\in R. For e_1 \gg e_2 \gg \ldots \gg e_{n-1} \gg e_ n = 1 the polynomial
g(x_1 + x_ n^{e_1}, x_2 + x_ n^{e_2}, \ldots , x_{n - 1} + x_ n^{e_{n - 1}}, x_ n) = ax_ n^ d + \text{lower order terms in }x_ n
where d > 0 and a \in R is one of the nonzero coefficients of g.
Proof.
Write g = \sum _{\nu \in N} a_\nu x^\nu with a_\nu \in R not zero. Here N is a finite set of multi-indices as in Lemma 10.115.1 and x^\nu = x_1^{\nu _1} \ldots x_ n^{\nu _ n}. Note that the leading term in
(x_1 + x_ n^{e_1})^{\nu _1} \ldots (x_{n-1} + x_ n^{e_{n-1}})^{\nu _{n-1}} x_ n^{\nu _ n} \quad \text{is}\quad x_ n^{e_1\nu _1 + \ldots + e_{n-1}\nu _{n-1} + \nu _ n}.
Hence the lemma follows from Lemma 10.115.1 which guarantees that there is exactly one nonzero term a_\nu x^\nu of g which gives rise to the leading term of g(x_1 + x_ n^{e_1}, x_2 + x_ n^{e_2}, \ldots , x_{n - 1} + x_ n^{e_{n - 1}}, x_ n), i.e., a = a_\nu for the unique \nu \in N such that e \cdot \nu is maximal.
\square
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