Multiplicity of a hypersurface equation at a rational point

Translate coordinates by $t_i=X_i-a_i$: the substitution $t_i\mapsto X_i-a_i$ is a polynomial-ring isomorphism with inverse $X_i\mapsto t_i+a_i$, and evaluation at $a$ becomes evaluation at the origin. By [F1], the latter has kernel generated by the variables, since every monomial with zero constant term is divisible by some $t_i$; hence its transported kernel is ${\mathfrak{m}}_a=(X_1-a_1,\dots ,X_n-a_n)$. Thus $P\mathrm{/}{\mathfrak{m}}_a\cong k$, so [F4] makes ${\mathfrak{m}}_a$ maximal and [F11] makes it prime; [F3] and [F5] identify $R$ as a local ring with maximal ideal $\mathfrak{m}$. Evaluation extends to $R$ because every denominator outside ${\mathfrak{m}}_a$ has nonzero value at $a$, and it identifies $R\mathrm{/}\mathfrak{m}$ with $k$. The affine stalk identification [F10] gives $R={\mathcal{O}}_{{\mathbb{A}}_k^n,a}$; translation identifies this filtered local ring with $k[t_1,\dots ,t_n{]}_{(t_1,\dots ,t_n)}$.

Write $f(a+t)={\sum }_j{f}_j(t)$ as in [F1]–[F2], and let $d$ be the least index with $f_d\mathrm{\ne }0$; it exists because translation is an isomorphism and $f\mathrm{\ne }0$, and $d\ge 1$ because $f(a)=0$. Apply [F6] with $I=0$ at the rational origin: it identifies the associated graded local ring with $k[t_1,\dots ,t_n]$, taking the degree-$j$ symbol of a polynomial to its degree-$j$ homogeneous part. Hence $f\in {\mathfrak{m}}^d$ and its class in ${\mathfrak{m}}^d\mathrm{/}{\mathfrak{m}}^{d+1}$ is the nonzero polynomial $f_d$, so $f\mathrm{\notin }{\mathfrak{m}}^{d+1}$ and its $\mathfrak{m}$-adic order is exactly the least degree $d$.

Let $u\in R^{\times }$. Its degree-zero initial class is its nonzero residue in $R\mathrm{/}\mathfrak{m}\cong k$, while the degree-$d$ initial class of $f$ is $f_d\mathrm{\ne }0$ by step 2.1. By [F7], the initial class of $uf$ is the product of these classes; [F8]–[F9] make the identified graded ring $k[t_1,\dots ,t_n]$ a domain, so the product is nonzero. Thus $uf\in {\mathfrak{m}}^d\setminus {\mathfrak{m}}^{d+1}$ and ${\mathrm{ord}}_{\mathfrak{m}}(uf)=d={\mathrm{mult}}_a(f)$. For $r\ge 1$, the initial class of $f^r$ is $f_d^r$, nonzero and homogeneous of degree $rd$ in the same domain, so ${\mathrm{mult}}_a(f^r)=rd$.

In one variable, $f=(X-a{)}^3+(X-a{)}^4$ has multiplicity $3$, unchanged after multiplication by the local unit $1+(X-a)$ by step 3.1; $X-a$ has multiplicity $1$. Degree zero cannot occur for a nonzero polynomial vanishing at $a$, and the zero polynomial is excluded because it has no least nonzero homogeneous part. If $n=0$, every polynomial is constant, so no nonzero polynomial vanishes at the unique point of $k^0$. The proof uses only the fixed coordinate translation, the unique finite polynomial expansion, and a fixed local unit; it makes no family selection and uses neither AC nor DC. There is no interval or endpoint parameter and no iff assertion in this definition.