= Magnus degree of a free pro-p group element
{c}
{title2=$D(w)=\deg(\iota(w)-1)$}
Embed a <free pro-p group> on $x_1,\ldots,x_d$ into the units of the noncommutative power-series algebra $\mathbb F_p\langle\!\langle X_1,\ldots,X_d\rangle\!\rangle$ by $x_i\mapsto1+X_i$. Its Magnus degree is the lowest total degree occurring in $\iota(w)-1$, with $D(1)=\infty$. Since $(1+v)^{p^e}=1+v^{p^e}$ in characteristic $p$, one has $D(w^{p^e})\ge p^eD(w)$. High powers therefore impose relations with small weight at any fixed parameter $0<\tau<1$.
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