Multiplicity one is the smooth hypersurface test

Translate to $t_i=X_i-a_i$ and write $f(a+t)=L(t)+$ terms of degree at least two. By [F1], ${\mathrm{mult}}_a(f)=1$ exactly when $L\mathrm{\ne }0$. For each $i$, the coefficient of $t_i$ in $L$ is $\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{X}_i(a)$: expanding each monomial $(a_j+t_j{)}^{e_j}$, its linear $t_i$ coefficient is the formal derivative coefficient, with the integer exponent interpreted in $k$. Thus multiplicity one is equivalent, in every characteristic, to at least one partial derivative being nonzero at $a$.

Suppose ${\mathrm{mult}}_a(f)=1$, and choose the least index $i$ with $h=(\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{X}_i)(a)\mathrm{\ne }0$. The class of the polynomial $\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{X}_i$ is outside $\mathfrak{q}$ in $A$, so localizing there gives the one-equation presentation $k[X_1,\dots ,X_n]\mathrm{/}(f)$ with its $1\times 1$ Jacobian minor invertible. Since $n\ge 1$, this is a standard smooth chart by [F2]; hence the structure morphism is smooth at $x$ by [F3].

Conversely, suppose the structure morphism is smooth at $x$. By [F2]–[F3], shrink once around $x$ to a finite-presentation standard smooth chart $B$ and let ${\mathfrak{q}}_B$ be its prime for $x$. The local $k$-module $B_{{\mathfrak{q}}_B}$ is flat by [F6]. Apply [F4] with base field $k$ and its zero prime: the fiber is $B$, and it is geometrically regular at ${\mathfrak{q}}_B$. Taking the extension $k\mathrm{/}k$ in [F5] shows that $B_{{\mathfrak{q}}_B}$ is regular. The chart does not change the stalk, so [F11] gives $B_{{\mathfrak{q}}_B}\cong A_{\mathfrak{q}}$; then [F10] identifies this ring with $R\mathrm{/}(f)R$. By [F9], $R$ is regular local, and $J=(f)R\subseteq \mathfrak{m}$ is nonzero by the finite order in [F1]. By [F7], $J$ is generated by an initial part $u_1,\dots ,u_r$ of a regular system of parameters. Their classes in $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ are linearly independent by [F8]. Since $J=(f)R$ and $f\in \mathfrak{m}$, every $sf$ modulo ${\mathfrak{m}}^2$ is the residue of $s$ times the class of $f$, so the image of $J$ in $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ has dimension at most one. Hence $r\le 1$. If $r=0$, [F7] makes $J=0$, contrary to [F1]; therefore $r=1$, and the image of $J$ is nonzero. It is spanned by the class of $f$, so $f\mathrm{\notin }{\mathfrak{m}}^2$. By [F1], ${\mathrm{mult}}_a(f)=1$.

In one variable, at $a=0$, $f=t$ has multiplicity $1$ and derivative $1$, giving the standard smooth chart; $f=t^p$ in characteristic $p>0$ has multiplicity $p$ and derivative $0$, and step 2.2 rules out smoothness. The zero polynomial is excluded because it has no least nonzero homogeneous part; a zero-variable polynomial cannot meet the nonconstant hypothesis. If the hypersurface has no $k$-rational points there is no instance of this pointwise claim, and there is no interval or endpoint parameter. Steps 2.1 and 2.2 establish both iff directions. The coefficient and chart arguments make no family of choices; AC is used through [F4], [F6], [F7], [F8], and [F9], and no additional DC assumption is used.