Subanalytic sets and limiting tangent directions

At a smooth point, a one-form can be tested on the tangent space. At a singular point, an approach to the set may have zero first derivative, and the first useful displacement may occur several orders later. Subanalytic geometry lets us replace such a limiting approach by an analytic curve and retain its first nonzero term. This gives a precise meaning to a one-form vanishing on a singular set, and explains why that condition survives analytic maps.

We work with finite-dimensional real analytic manifolds, Hausdorff and countable at infinity. All maps called analytic are real analytic. No coefficient ring or sheaf boundedness condition enters this geometric lesson. Later, these results will control cotangent supports of constructible sheaves.

The proof has three routes. Analytic curves turn limiting displacements into leading Taylor coefficients; proper uniformization lets tangent vectors detect a one-form; ordered radial graphs produce compatible triangulations. These routes share subanalytic set calculus but use different additional inputs. The signed normal-deformation criterion comes from the normal-geometry course.

Original exposition by GPT-6.1 Sol (OpenAI), Ultra, September–October 2026; revised by GPT-6 Astra (OpenAI), Ultra, October 2026. Original programme expression is public domain (CC0). Human results retain their named credit, and the linked Valette adaptations retain CC BY 4.0.

A local class with controlled boundaries

A subset $S\subset X$ is subanalytic if, near every point of the ambient manifold, it has a presentation

$$S\cap U =U\cap\bigcup_{j=1}^N \left(f_j^1(Y_j^1)\setminus f_j^2(Y_j^2)\right), \qquad\text{(1)}$$

where the $Y_j^i$ are compact real analytic manifolds and the $f_j^i:Y_j^i\to X$ are analytic. Both the neighborhood $U$ and the finite presentation can depend on the ambient point. In particular, behavior at a boundary point outside $S$ still matters.

An equivalent description uses local projections of relatively compact semianalytic sets. A semianalytic set is locally a finite Boolean combination of analytic equalities and strict inequalities. This description makes analytic rank conditions, specified by minors of a differential, available inside the subanalytic calculus. The compact-image construction below proves the equivalence with (1), including the converse for arbitrary subsets by a terminating sequence of frontier differences.

The following six statements fix the scope of the lesson. The compact-image construction proves the set and map calculus in item 1. The dimension-controlled proper uniformization proves item 4 on arbitrary noncompact manifolds as well, with a source of the same dimension as the set. The analytic-descent and squared-distance argument proves every fixed-dimensional intrinsic regular-locus statement in item 2. Curve selection is connected below to its explicit local-chart and Puiseux providers, and compatible triangulation is constructed first on a relative polyhedron and then on an arbitrary manifold. The analytic-function resolution in item 5 is proved below by constructing the local invariant, closed canonical centres and finite local towers, and then comparing and gluing those towers. The proof retains the entire function and its multiplicities.

1. Set operations and analytic maps. Closure, interior, finite union, intersection and difference preserve subanalyticity. Connected components are subanalytic and form a locally finite family in the ambient manifold. Analytic inverse images are subanalytic. If $f:Y\to X$ is analytic, $W\subset Y$ is subanalytic, and $f|_{\overline W}:\overline W\to X$ is proper, then $f(W)$ is subanalytic. The bar in this properness hypothesis is essential.
2. Regular points. A point of $S$ is regular when $S\cap U$ is a closed analytic submanifold of some ambient neighborhood $U$. Write $S_{\mathrm{reg}}$ for these points. The regular locus and its complement in $S$ are subanalytic; $S\subset\overline{S_{\mathrm{reg}}}$. For each fixed dimension $d$, the dimension-$d$ part of the regular locus is subanalytic. We set $\dim S=\sup_{x\in S_{\mathrm{reg}}}\dim_x S$, with the usual empty-set convention. The intrinsic regular-locus proof below establishes these statements using analytic descent and squared distance, with the Bierstone–Milman source credited there.
3. Curve selection. If $x\in\overline S$, there is an analytic curve $a:(-1,1)\to X$ with $a(0)=x$ and $a(t)\in S$ for every $t\ne0$. Its first derivative may vanish. The two-sided parameter is compatible with a one-sided set: squaring an initially positive parameter produces it.
4. Uniformization. A closed subanalytic set $S\subset X$ is the image of an analytic manifold $Y$ under a proper analytic map $f:Y\to X$. For nonempty $S$, the source can have pure dimension $\dim S$; if $S$ is compact, the source can be compact.
5. Resolution of an analytic function. Suppose $\varphi:X\to\mathbb R$ is analytic and is not identically zero on any connected component. Put $Z=\{\varphi=0,d\varphi=0\}$. There is a proper analytic $f:Y\to X$, an isomorphism over $X\setminus Z$, such that near each point of $f^{-1}Z$, in analytic coordinates, $$\varphi\circ f=\pm\prod_i y_i^{r_i},\qquad r_i\in\mathbb Z_{\ge0}. \qquad\text{(2)}$$ Exponents equal to zero are allowed.
6. Compatible triangulation. Every locally finite subanalytic partition of $X$ admits a locally finite simplicial triangulation whose open simplices have subanalytic analytic-submanifold images, each lying in one member of the partition. The triangulating map is a homeomorphism. Its global analyticity is not asserted. This is the bridge to Constructible sheaves on a triangulation.

Local finiteness extends the finite set-operation statements to a locally finite union: near any ambient point, only finitely many members occur. It does not authorize arbitrary unions. Likewise, analytic maps need not preserve subanalyticity of their images without the stated control at infinity.

Analytic coordinates and uniqueness with parameters

The coordinate arguments here require analytic charts and inverses. Their starting point is the proved holomorphic inverse and implicit function theorem, Lemma 2.2. The proof uses a uniformly contracting iteration on a small complex ball; its iterates are holomorphic, and Cauchy’s formula makes their locally uniform limit holomorphic. We record the passage to real analytic maps, including parameters, and the differential-equation fact needed by the finite-jet construction.

Real analytic inverse, implicit and constant-rank coordinates

A real analytic power series, on a sufficiently small neighborhood of its centre, extends to a holomorphic power series on a complex polydisc. Indeed, absolute convergence on a smaller real box bounds the series with the absolute values of its coefficients; the same bound applies to complex arguments of those moduli. Its coefficients are real, so this extension commutes with conjugation.

Analytic inverse theorem with parameters. Let $F(x,\lambda)$ be real analytic near $(a,\lambda_0)$, with $x\in\mathbb R^m$, $\lambda\in\mathbb R^p$, and assume $D_xF(a,\lambda_0)$ is invertible. There is a unique local real analytic map $G(y,\lambda)$ solving $$F(G(y,\lambda),\lambda)=y, \qquad G(F(x,\lambda),\lambda)=x. \qquad\text{(LAF1)}$$

Proof. Complexify $F$ near the specified point. The block map $$(x,\lambda)\longmapsto(F(x,\lambda),\lambda) \qquad\text{(LAF2)}$$ has invertible derivative there. Apply the holomorphic inverse theorem. Its inverse preserves the last coordinate because the map itself does. After taking conjugation-invariant neighborhoods, uniqueness implies that the inverse commutes with conjugation: conjugating the unique preimage of a real target gives that same preimage. Restriction to real points therefore gives a real analytic inverse. The convergent complex power series of its components restrict to the required real power series. Both identities follow from the two inverse identities. $\square$

Consequently, if $H(x,z,\lambda)$ is real analytic, $H(a,b,\lambda_0)=0$, and $D_zH(a,b,\lambda_0)$ is invertible, the map $$(x,z,\lambda)\longmapsto(x,H(x,z,\lambda),\lambda)$$ has a real analytic inverse. Its middle target coordinate equal to zero gives a unique real analytic solution $z=h(x,\lambda)$. More generally, independent defining differentials can be completed by linear coordinate functions to an invertible square Jacobian. This gives analytic submersion coordinates, analytic local sections, and analytic level submanifolds, with tangent space the kernel of the differential.

Analytic constant-rank theorem with parameters. Suppose $F(x,\lambda)\in\mathbb R^n$ is real analytic and $$\operatorname{rank}D_xF(x,\lambda)=r \qquad\text{(LAF3)}$$ throughout a neighborhood of $(a,\lambda_0)$. There are real analytic source and target coordinate changes, both preserving $\lambda$, in which the map becomes $$(u,z,\lambda)\longmapsto(u,0,\lambda). \qquad\text{(LAF4)}$$ The hypothesis is constant rank in the $x$ variables on a neighborhood, not merely a rank calculation at the central point.

Proof. For $r>0$, reorder source and target coordinates so an $r$-by-$r$ minor is invertible at the specified point. The map $$(x,\lambda)\longmapsto (F_1(x,\lambda),\ldots,F_r(x,\lambda), x_{r+1},\ldots,x_m,\lambda)$$ has invertible derivative and is an analytic coordinate change by (LAF1). On a smaller product box, the transformed map has the form $$(u,z,\lambda)\longmapsto(u,K(u,z,\lambda),\lambda).$$ At each fixed parameter its derivative in $(u,z)$ is $$\begin{pmatrix}I_r&0\\ D_uK&D_zK\end{pmatrix}.$$ Its rank equals $r+\operatorname{rank}D_zK$, by subtracting $D_uK$ times the upper block rows from the lower block rows. Condition (LAF3) forces $D_zK=0$. Integrating each derivative along coordinate segments inside the product box shows $K(u,z,\lambda)=k(u,\lambda)$. The analytic target change $$(u,w,\lambda)\longmapsto(u,w-k(u,\lambda),\lambda)$$ has the explicit analytic inverse which adds $k$, and gives (LAF4). For $r=0$, the same segment argument makes $F$ independent of $x$, and subtraction of this parameter-dependent value finishes the proof. $\square$

Taking no parameters gives the analytic constant-rank theorem on manifolds by using charts. It proves, in particular, that a local constant-rank fibre is an analytic manifold with tangent space $\ker dF$. A map with injective differential is locally an analytic embedding. If it is also a homeomorphism onto its image, these local descriptions give an embedded analytic submanifold globally on that image.

Analytic differential equations

For the finite-jet argument, uniqueness is already supplied by Local tools for bundles and transport, Theorem 2.1. That theorem proves uniqueness among differentiable solutions and smooth dependence on all initial data. Analytic dependence is a stronger assertion; the following direct construction establishes it when the vector field is analytic.

Analytic parameter theorem. For a real analytic finite system $$\dot x=V(t,x,\lambda),\qquad x(t_0)=a, \qquad\text{(LAF5)}$$ the local solution is real analytic jointly in $t,t_0,a,\lambda$. It is the unique differentiable real solution with those data on any common connected interval where the solutions are defined.

Proof. Translate a fixed datum to the origin and extend $V$ holomorphically to a complex product neighborhood. Use the maximum norm on coordinates and its induced operator norm for matrices. Choose positive $T,R,P$ whose closed product polydisc is inside that neighborhood, and bounds $M,L$ for $\lVert V\rVert$ and $\lVert D_xV\rVert$ there. The spatial segment formula gives $$\lVert V(t,x,\lambda)-V(t,y,\lambda)\rVert \le L\lVert x-y\rVert$$ inside the convex spatial polydisc. Choose $\rho>0$ so $$\rho<T/4,\qquad \rho M<R/4,\qquad q=\rho L<1/2. \qquad\text{(LAF6)}$$ Restrict $|t_0|<T/4$, $\lVert a\rVert<R/4$, and $\lVert\lambda\rVert<P/2$. If a bound is zero, its corresponding inequality imposes no restriction.

Starting with $u_0(s,a,t_0,\lambda)=a$, define, for $|s|<\rho$, $$u_{k+1}(s,a,t_0,\lambda) =a+\int_0^s V(t_0+\zeta,u_k(\zeta,a,t_0,\lambda),\lambda)\,d\zeta. \qquad\text{(LAF7)}$$ The integral is over the straight complex segment. Every iterate has spatial norm at most $R/2$: its norm is at most $R/4+\rho M<R/2$ whenever the preceding iterate has that bound. Its time argument stays inside $|t|<T/2$.

Every iterate is jointly holomorphic. To verify the integration step, write the integral as $s\int_0^1 V(t_0+\theta s,u_k(\theta s,a,t_0,\lambda),\lambda)\,d\theta$. On each smaller closed parameter polydisc the integrand is uniformly continuous in $\theta$; the Riemann sums are holomorphic and converge uniformly. The holomorphic convergence theorem, Theorem 3.1, makes the integral holomorphic.

Take suprema over the entire smaller data product and $|s|<\rho$. The spatial estimate gives $$\lVert u_{k+1}-u_k\rVert_\infty \le q\,\lVert u_k-u_{k-1}\rVert_\infty, \qquad \lVert u_1-u_0\rVert_\infty\le\rho M.$$ The geometric tail bound proves uniform convergence to a jointly holomorphic $u$. Uniform convergence of the integrands passes (LAF7) to the limit. For fixed data the integrand is holomorphic in $s$, so its integral is a primitive; consequently $\partial_su=V(t_0+s,u,\lambda)$ and $u(0)=a$. Substitution $s=t-t_0$ gives joint holomorphy in the stated variables on $|t-t_0|<\rho$.

All real data and real $s$ give real iterates, because the vector field’s extension agrees with the real vector field there. Hence the limit restricts to a jointly real analytic solution. If two differentiable real solutions have the same datum, restrict to a common small interval on which both remain in the spatial polydisc. Their integral equations give a supremum difference at most $L\epsilon$ times itself; choose $L\epsilon<1$. They agree there. The set of agreement times on a common connected interval is closed by continuity and open by this local argument applied at each agreement point. It is the whole interval. $\square$

Uniqueness gives the local composition law for solution maps: $$\Phi_\lambda(t,s,\Phi_\lambda(s,t_0,a)) =\Phi_\lambda(t,t_0,a). \qquad\text{(LAF8)}$$ Thus the time-$t_0$-to-$t$ map is an analytic local diffeomorphism, with inverse the time-$t$-to-$t_0$ map. These assertions hold on the neighborhoods where the indicated solutions exist. Along an already existing solution on a compact time interval, finitely many local constructions compose to give joint analytic dependence near that trajectory. No existence for all real time is inferred.

For the line-incidence construction, put $g_j=L^jh$, where $L=v\cdot\partial_x$. A local identity $g_{q+1}=\sum_{j=0}^q a_j(x,v)g_j$ gives, along $x+tv$, the system $$U_j'=U_{j+1}\ (j<q),\qquad U_q'=\sum_{j=0}^q a_j(x+tv,v)U_j. \qquad\text{(LAF9)}$$ Choose a smaller parameter box and a common time interval whose line segments remain inside the coefficient neighborhood. Its coefficient matrix has a common bound $B$. When $U(0)=0$, integration gives $\sup_{|t|\le\epsilon}\lVert U(t)\rVert \le B\epsilon\sup_{|t|\le\epsilon}\lVert U(t)\rVert$. For $B\epsilon<1$, $U=0$. This proves the exact zero-solution uniqueness used there, uniformly on that smaller parameter box. It does not require the stronger analytic parameter theorem.

A parameter-preserving normal form

For $$F_\lambda(x,y)=(x+y^2+\lambda y,\ (x+y^2+\lambda y)^2),$$ the spatial derivative has rank one everywhere. Set $u=x+y^2+\lambda y,\ z=y$ in the source and $(a,b)\mapsto(a,b-a^2)$ in the target. The inverse source change is $x=u-z^2-\lambda z,\ y=z$. Both changes preserve $\lambda$ and are analytic, and the transformed map is exactly $(u,z)\mapsto(u,0)$. This illustrates why a parameter-dependent coordinate change is allowed while the parameter itself remains fixed.

The example uses the source coordinates and target change displayed above. At the fixed parameter shown, each coloured fibre maps to its matching point on the parabola; subtraction of the square of the first coordinate puts the image on a straight line. The coordinate calculation proves the identity for every parameter, not only the plotted one. Full-size diagram · Reproducible Python source.

From a local analytic presentation to analytic curve selection

The human treatment of curve selection and Łojasiewicz inequalities, adapted from Guillaume Valette’s 2025 survey under CC BY 4.0, gives its arguments in globally subanalytic Euclidean spaces. Its cell-decomposition and Puiseux prerequisites are stated explicitly. The following chart argument explains how its curve-selection proof applies to the local sets in this lesson.

The companion analytic finiteness treatment now proves a prerequisite for that cell construction: real analytic Noetherianity via the existing Demailly division provider, Artin–Rees and Krull intersection, convergent replacements of formal linear solutions with prescribed jets, a finite Taylor family with analytic units, and the signed uniform-order normalization. The full analytic-composition and inverse-power reduction remains distinct from this finiteness proof.

Local comparison. Given $x\in X$ and a subanalytic $S\subset X$ with presentation (1) near $x$, there is an analytic coordinate neighborhood and a smaller bounded coordinate ball $B$ about $x$ such that the coordinate image of $S\cap B$ is globally subanalytic.

Proof. Choose the ball with compact closure inside both the presentation neighborhood and an analytic chart $V$. For one of the finitely many maps $f:Y\to X$ in (1), the set $K=f^{-1}(\overline B)$ is compact because $Y$ is compact. Cover $K$ by finitely many closed coordinate boxes $Q_\ell$ whose interiors cover $K$ and whose neighborhoods lie in $f^{-1}(V)$. This follows by first choosing such a box around each point of $K$ and then taking a finite subcover. In their coordinates, each $Q_\ell$ is a compact rectangle and the coordinate functions of $f$ are analytic on a neighborhood of that rectangle.

The graph of $f|_{Q_\ell}$ in these coordinates is bounded and semianalytic in its whole ambient Euclidean space: on a neighborhood of the box it is given by the rectangle inequalities and analytic graph equations; away from the closed box it is locally empty. A bounded ambient semianalytic set is globally semianalytic. Indeed, compactification $z\mapsto z/\sqrt{1+|z|^2}$ is an analytic coordinate change on a neighborhood of its compact closure, whose image stays strictly inside the unit ball. The compactified set is therefore semianalytic, and is empty near the compactification boundary. In the projective-product convention of the preparation theorem, boundedness makes the compact closure lie entirely in the finite affine chart; the set is empty near every point at infinity, so is globally semianalytic there as well.

Projecting this graph proves that $f(Q_\ell)$ is globally subanalytic. Every point of $f(Y)\cap B$ has a preimage in $K$, so

$$f(Y)\cap B=\left(\bigcup_\ell f(Q_\ell)\right)\cap B.$$

This is a finite union and intersection of globally subanalytic sets. Apply the same argument to every map in (1), and then apply the globally subanalytic Boolean-operation theorem to its finite differences and unions. The result is precisely $S\cap B$. $\square$

The full two-sided curve. Suppose $x\in\overline S$. Apply the cited component’s proved choice-and-Puiseux argument to the bounded coordinate set $S\cap B$. It gives an arc $\gamma:[0,\epsilon)\to S\cap B$ with endpoint $x$, analytic across zero. Choose $0<c<\epsilon$ small enough that the power series and chart inverse are defined whenever $|r|<c$. The analytic map

$$a(t)=\gamma(c t^2),\qquad -1<t<1,$$

interpreted in the analytic chart, has $a(0)=x$ and $a(t)\in S$ for every $t\ne0$. If $x\notin S$, it is nonconstant. This proves the exact two-sided statement from the local comparison and the explicit cell/Puiseux inputs; it imposes no global definability assumption on $S$ or $X$.

No assertion of finiteness at infinity has entered this argument. For instance, $\{(u,\sin u):u\in\mathbb R\}$ is analytic locally but cannot be globally subanalytic: intersecting with the horizontal axis and projecting would make its infinite discrete set of zeroes a definable subset of the line, contrary to finite cell decomposition. Each bounded chart piece nevertheless has the local comparison just proved. The ambient local finiteness and properness qualifications elsewhere in this lesson therefore remain in force.

Removing the unit from a resolved function

A resolution is often expressed with a nonvanishing analytic factor: $\varphi\circ f=u(y)\prod_i y_i^{r_i}$. The precise sign-only expression (2) follows from that form near a point of the zero set. This last coordinate step changes neither the map $f$ nor its isomorphism over $X\setminus Z$.

Unit normalization. Suppose a nonzero analytic function has the displayed form near a point $a$ where its value is zero. There are analytic coordinates centred at $a$ in which it is a sign times a monomial. The coordinate change preserves the normal-crossing coordinate hyperplanes through $a$.

Proof. Any factor with $y_i(a)\ne0$ is a local analytic unit; absorb it into $u$. Centre the remaining coordinates at $a$. At least one active exponent is positive, since the function vanishes there. Relabel it $r_1>0$. Shrink so that $u$ has constant sign $\varepsilon\in\{1,-1\}$. The positive function $|u|=\varepsilon u$ has a positive analytic $r_1$-th root $b$. Indeed, the implicit function theorem applied to $b^{r_1}-|u(y)|=0$ has nonzero derivative in $b$ at its positive root. Set

$$z_1=y_1b(y),\qquad z_i=y_i\ (i>1).$$

The Jacobian determinant at the centre is $b(a)>0$; all extra derivatives in the first row carry the factor $y_1(a)=0$. The analytic inverse function theorem supplies a coordinate change. Substitution gives $\varphi\circ f=\varepsilon\prod_i z_i^{r_i}$. Because $b$ never vanishes, $\{z_1=0\}=\{y_1=0\}$, and the other active hyperplanes are unchanged. This proves every claim. $\square$

The zero-set qualification matters. A nonconstant unit such as $e^x$ has no positive vanishing exponent to absorb it and cannot become the constant monomial $1$ by an invertible coordinate change. In the resolution input, points of $f^{-1}Z$ lie above $\{\varphi=0\}$, so the needed qualification holds. This coordinate argument supplies the last step from a unit-monomial resolution, not the construction of a singular-locus-preserving resolution itself.

Reducing marked equations to their coefficients

The source for the following coefficient construction is Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, author manuscript of 25 November 1996, Proposition 4.12, Construction 4.18 and Proposition 4.19, with the proofs on pp. 33–34. The edition cited here is the complete 70-page author manuscript. The argument uses a distinguished derivative to select a hypersurface and transfers each marked order to finitely many coefficients on it. Ordinary permissible blowings-up divide by the exceptional coordinate with the assigned weight; exceptional tests pull back without that division. Keeping these two transformation rules separate is essential to persistence.

Our real analytic proof uses a convergent implicit coordinate change, then verifies coefficient divisibility along the whole smooth centre, the ordinary charts, the omitted transverse direction and the two other test transformations. Its hypothesis that the transverse direction is compatible with the existing exceptional hypersurfaces is exactly the additional hypothesis needed in the source’s Proposition 4.12. The conclusion is a local reduction with finite persistence under those hypotheses. The residual invariant, exceptional history and choice of centres in the surrounding desingularization construction require further arguments, identified after the proof.

Let $U$ be an analytic coordinate neighborhood, with coordinates $(x,t)\in\mathbb R^{m-1}\times\mathbb R$, and let $\mathcal F=\{(h,b_h)\}$ be a finite nonempty collection of analytic functions with positive integer marks. Its equimultiple locus is

$$S_{\mathcal F}=\{p\in U:\operatorname{ord}_p h\ge b_h\text{ for every }(h,b_h)\in\mathcal F\}.$$

The order of the zero germ is infinite. Assume that at a chosen $a\in S_{\mathcal F}$, one distinguished pair $(h_*,d)$ satisfies $\operatorname{ord}_a h_*=d$ and $\partial_t^d h_*(a)\ne0$. If exceptional hypersurfaces are present, assume they are coordinate hypersurfaces in the $x$ variables. Thus the distinguished transverse direction is compatible with them. This is an explicit hypothesis when exceptional divisors have already appeared.

Coefficient reduction and persistence. After an analytic change of the transverse coordinate, there is a smooth hypersurface $N=\{t=0\}$, transverse to these exceptional hypersurfaces, and expansions

$$h(x,t)=\sum_{q=0}^{b_h-1}c_{h,q}(x)t^q+t^{b_h}R_h(x,t).$$

Here $c_{*,d-1}=0$ and $R_*(a)\ne0$. In a neighborhood of $a$,

$$S_{\mathcal F} =\{p\in N:\operatorname{ord}_p c_{h,q}\ge b_h-q \text{ for every }h,\ 0\le q<b_h\}.$$

The orders on the right are computed on $N$. If a smooth centre $C\subset S_{\mathcal F}$ has normal crossings with the exceptional hypersurfaces and $N$, this equality persists near every point above $a$ where the transformed marked equations still have their marked orders. In a chart with exceptional coordinate $y_k$, the coefficients transform with their own marks:

$$h'=y_k^{-b_h}(h\circ\sigma),\qquad c'_{h,q}=y_k^{-(b_h-q)}(c_{h,q}\circ\sigma_N).$$

The hypersurface $N$ transforms strictly, with local equation $t'=t/y_k$. Its transforms remain smooth and have normal crossings with the transformed exceptional hypersurfaces. The same coefficient reduction also persists under adjoining an independent variable and under an exceptional blowing-up in two existing exceptional hypersurfaces. For the latter test transformation the functions are pulled back without dividing by an exceptional power. Repeated applications give the corresponding assertion along every finite sequence of these transformations, at points satisfying the marked order conditions.

Proof of the initial reduction. Put $z=\partial_t^{d-1}h_*$. Its derivative with respect to $t$ is nonzero at $a$, so the analytic implicit function theorem gives $z=0$ as a graph $t=\psi(x)$. Replace $t$ by $t-\psi(x)$, leaving $x$ unchanged. Shrinking the neighborhood, $z=t\,u(x,t)$ with $u$ an analytic unit. Differentiation in the new $t$ coordinate is the same as differentiation in the old one with $x$ held fixed. The exceptional coordinate equations are unchanged.

Taylor expansion in $t$, with analytic remainder, gives the displayed expansions. Since $z(x,0)=0$, $c_{*,d-1}=0$. The coefficient $R_*(a)=\partial_t^d h_*(a)/d!$ is nonzero. If the order of $h_*$ at a nearby point is at least $d$, its derivative $z$ vanishes there. Thus every point of $S_{\mathcal F}$ lies on $N$.

At $p=(x_0,0)$, expand each coefficient in $x-x_0$. Terms with distinct powers $t^q$ cannot cancel. The remainder $t^{b_h}R_h$ has total order at least $b_h$. Consequently,

$$\operatorname{ord}_{(x_0,0)}h\ge b_h \quad\Longleftrightarrow\quad \operatorname{ord}_{x_0}c_{h,q}\ge b_h-q \quad(0\le q<b_h).$$

This proves the equality of loci. The distinguished $t^d$ coefficient also shows that the order of $h_*$ at any point of this locus is exactly $d$.

Divisibility along a permissible centre. Straighten $C$ inside $N$, keeping $t$ as transverse coordinate and the exceptional equations as coordinate equations. Locally,

$$C=\{t=0,\ x_i=0\ (i\in J)\}.$$

Write $I_J=(x_i:i\in J)$ on $N$. The coefficient conditions hold at every point of $C$, not merely at $a$. Hence $c_{h,q}\in I_J^{b_h-q}$. To see this directly, expand a coefficient in the normal variables $x_i$, $i\in J$, with analytic coefficients in the tangential variables. Every normal derivative of total degree less than $b_h-q$ vanishes at every point of $C$, by the order condition there. All the corresponding tangential coefficient functions are therefore identically zero. This proves membership in the stated ideal power and makes every division in the transform formula analytic.

The ordinary blow-up charts. For $k\in J$, the $x_k$ chart is

$$x_k=y_k,\qquad x_i=y_ky_i\ (i\in J\setminus\{k\}),\qquad t=y_ks,\qquad x_i=y_i\ (i\notin J).$$

Substitution and division give

$$h'(y,s)= \sum_{q<b_h}y_k^{-(b_h-q)}c_{h,q}(\sigma_N(y))s^q +s^{b_h}R_h(\sigma(y,s)).$$

The strict transform $N'$ is $s=0$. In the distinguished equation the coefficient of $s^{d-1}$ is zero. Near any point above $a$,

$$\partial_s^{d-1}h'_* =s\bigl(d!R_*(\sigma(y,s))+y_ks\,A(y,s)\bigr)$$

for an analytic $A$. The factor in parentheses restricts to the nonzero constant $d!R_*(a)$ on that exceptional fibre. It is a unit locally. The distinguished $d$-th derivative is likewise nonzero there. The initial coefficient argument therefore applies to $h'$: its equimultiple locus lies on $N'$ and equals the locus of the displayed coefficient conditions with marks $b_h-q$.

The remaining points above $a$ must be checked in the $t$ chart. Here $t=v$, $x_i=vw_i$ for $i\in J$, and the other $x_i$ are unchanged. A point in this chart that belongs to no $x_k$ chart has $w_i=0$ for every $i\in J$. Since $c_{*,q}\in I_J^{d-q}$, the quotient $v^{-(d-q)}c_{*,q}(vw,x_{\notin J})$ restricts to zero at such a point. The transformed $h_*$ therefore has value $R_*(a)\ne0$. Its order is zero, so that point is outside the transformed equimultiple locus. This covers every projective direction. If $J$ is empty, all lower coefficients vanish identically and this same $t$-chart argument removes the whole marked locus.

In the $x_k$ charts, the new exceptional divisor is $y_k=0$; each surviving old exceptional divisor is another coordinate hypersurface in the $y$ variables. Together with $s=0$ they have normal crossings. This proves the geometric assertion as well as the coefficient formula.

The two other test transformations. An independent product variable changes none of the expansions, orders or equations. For a blowing-up of two exceptional hypersurfaces, choose equations $x_1=0,x_2=0$. In the relevant chart the substitution is $x_1=y_1,x_2=y_1y_2$, with $t$ unchanged. The controlled test transform is now $h'=h\circ\sigma$, so $c'_{h,q}=c_{h,q}\circ\sigma_N$. Pullback cannot decrease the order of a coefficient at a point above $a$: substituting analytic germs vanishing at $a$ into every Taylor monomial preserves its minimum degree. The distinguished $t^d$ coefficient remains a unit and $c'_{*,d-1}=0$. Thus the same order-locus equality and the transverse coordinate equation $t=0$ hold. The other chart is identical with the two exceptional coordinates interchanged; the prescribed distinguished point of an exceptional test is included in these charts.

After any of the transformations, the data again satisfy the initial hypotheses at a point where all marked orders persist. The distinguished order is exactly $d$, the distinguished derivative is nonzero, and the coefficient expansions and exceptional coordinate equations have the required form. Applying the argument successively proves persistence for every finite allowed sequence. $\square$

This is the local dimension reduction: it replaces conditions on functions of $m$ variables by weighted conditions on functions of $m-1$ variables. The later sections prove recovery of divisor orders, the recursive history and coefficient construction, closed canonical centre selection, compact-local termination, and overlap compatibility. A multiplicity drop alone also need not give normal crossings with the exceptional divisor, as the cusp exercise below demonstrates.

Gluing canonical local resolutions into a proper global map

A function-resolution construction must fit across coordinate neighborhoods. The same Bierstone–Milman author manuscript, Theorem 13.2 and the proof of Theorem 13.3, passes from resolutions on relatively compact open sets to a global analytic map by unique compatible lifts. Its proof relies on the previously constructed invariant and local algorithm. Theorem 13.4 then treats the stronger requirement of a locally finite sequence of global blowings-up by extending and resolving centres.

The lemma below isolates the descent argument: the local proper maps and their isomorphisms over the full overlaps are hypotheses. Density forces uniqueness and the cocycle; quotient charts give the manifold; a finite compact-cover argument proves properness. This supplies the gluing step once compatibility exists, while leaving the construction of compatible local resolutions and the extension of centres to their separate proofs.

Gluing lemma. Let $X$ be a real analytic manifold with a countable locally finite relatively compact open cover $(U_i)$, and let $Z\subset X$ be closed. Suppose that each $U_i$ has a proper surjective analytic map $f_i:Y_i\to U_i$ from an analytic manifold, an analytic isomorphism over $U_i\setminus Z$, with $f_i^{-1}(U_i\setminus Z)$ dense in $Y_i$. Suppose also that for every overlap there is an analytic isomorphism

$$g_{ji}:f_i^{-1}(U_i\cap U_j)\longrightarrow f_j^{-1}(U_i\cap U_j), \qquad f_jg_{ji}=f_i.$$

Then these maps glue to an analytic manifold $Y$ and a proper surjective analytic map $f:Y\to X$, an analytic isomorphism over all of $X\setminus Z$. If an analytic function $\varphi$ pulls back under every $f_i$ to an analytic unit times a coordinate monomial near $f_i^{-1}Z$, the same holds for $f$.

Proof. The overlap isomorphisms are unique. On the dense open inverse image of $(U_i\cap U_j)\setminus Z$, any such isomorphism must be $f_j^{-1}f_i$. Two continuous maps to the Hausdorff manifold $Y_j$ that agree on this dense set agree everywhere: their equality set is closed, being the inverse image of the diagonal. Density restricts to every open overlap. In particular,

$$g_{ii}=\mathrm{id},\qquad g_{ij}g_{ji}=\mathrm{id},\qquad g_{kj}g_{ji}=g_{ki}$$

on their respective domains. The last equality follows by comparing both maps over a triple overlap. Thus the required cocycle is a consequence of uniqueness.

Take the disjoint union of the $Y_i$ and identify a point of $Y_i$ with its image under $g_{ji}$ whenever its base point belongs to the overlap. The cocycle shows that each $Y_i$ injects into the quotient, that its image is open, and that the induced topology and analytic charts there are its original ones. More precisely, the preimage over $U_i$ of this quotient is precisely $Y_i$: every representative over $U_i$ has a unique representative in that chart. The $f_i$ therefore descend to a map $f$ with $f^{-1}(U_i)\cong Y_i$.

The quotient is Hausdorff. If two points have different base points, disjoint base neighborhoods separate them. If they have the same base point, choose $U_i$ containing it; both points lie in the same open Hausdorff chart $Y_i$, which separates them by open subsets of the quotient. It is second countable because it is the union of countably many open second-countable $Y_i$; the union of their countable bases is a basis. The analytic transition maps now make $Y$ an analytic manifold. The map $f$ is analytic and surjective by its local descriptions.

To check properness, let $C\subset X$ be compact. Choose finitely many open neighborhoods $V_1,\ldots,V_m$ covering $C$, with each compact closure $\overline V_\ell$ contained in some $U_{i(\ell)}$. Such neighborhoods exist by local compactness and regularity of a manifold. Each $C\cap\overline V_\ell$ is a compact subset of that $U_{i(\ell)}$. Hence

$$f^{-1}(C)=\bigcup_{\ell=1}^{m} f_{i(\ell)}^{-1}(C\cap\overline V_\ell)$$

is a finite union of compact subsets of $Y$. This proves properness. Merely gluing maps that are proper over smaller pieces, without control over their full inverse images on overlaps, would not justify this step.

Over $X\setminus Z$, the local inverses $f_i^{-1}$ agree on overlaps, so they give a global analytic inverse. The coordinate monomial expressions are local statements in the charts $Y_i$, and therefore remain true on $Y$. If $\varphi=0$ on $Z$, the preceding unit-normalization lemma gives the exact sign-only expression near every point of $f^{-1}Z$, while preserving the map $f$ itself. This proves the lemma. $\square$

The density hypothesis makes the canonical transition unique. For a local construction by blowings-up, it is supplied by the dense complement of the exceptional sets in the coordinate charts. The later centre, termination and independent-tower comparison theorems construct the local maps and their full-overlap isomorphisms required here. The gluing proof does not turn the global map into a prescribed sequence of global blowings-up: that stronger construction has its own extension-of-centres argument.

Why the real zero set does not determine the resolved function

The function

$$\varphi(x,y)=xy(x^2+y^2)$$

has the two coordinate axes as its real zero set. They already cross normally. Nevertheless, $\varphi$ is not a unit times a coordinate monomial at their intersection, even after an invertible analytic coordinate change.

Proof. At a nonzero point of either axis, the other factors are units, so the order of vanishing along that branch is one. If a coordinate monomial had exactly these two branches, each exponent would consequently be one. Its order at their intersection would be two. But the displayed $\varphi$ has order four at the origin. The order of an analytic germ is unchanged by an analytic coordinate change: its first nonzero homogeneous term is composed with the invertible linear part of the change and remains nonzero in the same degree. Multiplication by an analytic unit does not change that degree either. Thus the two orders contradict the proposed monomial form. $\square$

The missing factor $x^2+y^2$ vanishes at the crossing, although it contributes no additional real branch. Resolving only the pictured real support loses this information. One real analytic blowing-up of the origin resolves this particular function:

$$(x,y)=(u,uv) \quad\Longrightarrow\quad \varphi=u^4v(1+v^2).$$

The factor $1+v^2$ is a positive analytic unit. The other chart gives the corresponding expression at the vertical direction. The following exercise proves the complete proper-map and isomorphism assertions, rather than treating one affine substitution as the whole resolution.

Keeping the whole function through a resolution

A resolution of a function retains its vanishing multiplicities. Resolving only the reduced real zero set does not supply this information. We first prove the complete-function calculation for a supplied tower of blowings-up, identify the centre condition which protects the original regular-zero locus, and carry out a complete cusp example. The canonical construction and termination theorems later in the lesson supply such towers in general.

The ideal-theoretic endpoint is treated in Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, complete author manuscript of 25 November 1996, Remark 1.8 and Theorem 1.10, pp. 7–8. The principal-ideal calculation below explains directly why an endpoint with unit weak ideal gives the required multiplicities of the actual function.

The total exponent belongs to the function

Let $M$ be a finite-dimensional Hausdorff second-countable real analytic manifold. Let $f:M\to\mathbb R$ be analytic and not identically zero on any connected component. A supplied admissible principal tower consists of finitely many real projective blowings-up

$$M_j\xrightarrow{\sigma_j}M_{j-1}\longrightarrow\cdots \xrightarrow{\sigma_1}M_0=M, \qquad \pi_j=\sigma_1\circ\cdots\circ\sigma_j:M_j\longrightarrow M, \qquad\text{(PF1)}$$

with $\pi_0=\operatorname{id}_M$, and the following data. The centre of $\sigma_{j+1}$ is a closed embedded smooth analytic submanifold $C_j\subset M_j$ of positive codimension. Disconnected centres are allowed; the conditions below are imposed locally on each component. Empty steps can be omitted.

There is a locally principal analytic ideal sheaf $I_j\subset\mathcal O_{M_j}$, with nonzero local generator $h_j$, and a locally finite labelled family $E_j$ of distinct smooth analytic hypersurfaces with simple normal crossings. Initially $I_0=(f)$ and $E_0=\varnothing$. For every $H\in E_j$, every point of $H$, and every local equation $z_H$ of $H$, the germ of $h_j$ is not divisible by $z_H$. Equivalently, its restriction to the germ of $H$ is not the zero analytic germ. It may still vanish at individual points of $H$.

The centre lies in $V(I_j)$. The point order $\operatorname{ord}_p h_j$ is a constant positive integer $m_j$ on each connected centre component; this is independent of the generator, since multiplication by an analytic unit preserves order. The centre and the old divisors admit simultaneous coordinates: locally the centre is a coordinate subspace, and the divisors through the point are distinct coordinate hyperplanes. This is the normal-crossing condition used below.

Put $F_{j+1}=\sigma_{j+1}^{-1}(C_j)$. Its ideal $I(F_{j+1})$ is invertible. Define

$$I_{j+1}=I(F_{j+1})^{-m_j} \bigl(I_j\mathcal O_{M_{j+1}}\bigr), \qquad h_{j+1}=e^{-m_j}(h_j\circ\sigma_{j+1}), \qquad\text{(PF2)}$$

where $e$ is a local generator of $I(F_{j+1})$. Here $I_j\mathcal O_{M_{j+1}}$ denotes the ideal generated by the analytic pullbacks of sections of $I_j$; the negative power in (PF2) is multiplication by the inverse invertible ideal. When the centre orders differ between components, the formula is read with their locally constant values. The chart calculation below proves that the result is an analytic ideal, rather than merely a fractional one. Changes of $h_j$ or $e$ multiply $h_{j+1}$ by analytic units.

The updated family $E_{j+1}$ consists of the nonempty strict transforms of the old hypersurfaces, retaining their labels, and the components of $F_{j+1}$, with new labels. The strict transform of $H$ is the closure of $\sigma_{j+1}^{-1}(H\setminus C_j)$. We also record $F_{j+1}$ when the centre is Cartier; that records a divisor factor even when the underlying blow-up is an isomorphism.

Total-transform lemma. Formula (PF2) gives nonzero analytic weak generators. They have no factor from any germ of the updated divisor family, and that family has simple normal crossings. Locally the actual function has the equality

$$f\circ\pi_j=u_jh_j\prod_{H\in E_j}z_H^{a_{H,j}}, \qquad a_{H,j}\in\mathbb Z_{\geq0}, \qquad\text{(PF3)}$$

where $u_j$ is an analytic unit and only the hypersurfaces meeting the coordinate neighbourhood are included. The exponents are locally constant along the corresponding labelled hypersurfaces. The old strict-transform exponents are unchanged. At a new divisor over a connected centre piece,

$$a_{F_{j+1},j+1} =m_j+\sum_{H\supset C_j}a_{H,j}. \qquad\text{(PF4)}$$

The containment condition in (PF4) means containment of the local centre germ. A divisor meeting the centre transversely, without containing it, contributes no exceptional factor.

Proof of analytic divisibility and exact exceptional order. Work in simultaneous coordinates $(x_1,\ldots,x_c,w)$, with $C_j=\{x=0\}$. A convergent expansion in the normal variables has the form

$$h_j(x,w)=\sum_\alpha A_\alpha(w)x^\alpha.$$

For $|\alpha|<m_j$, the normal derivative $\partial_x^\alpha h_j(0,w)$ vanishes at every nearby centre point, because the point order there is at least $m_j$. Thus $A_\alpha$ is identically zero. In particular,

$$h_j\in I(C_j)^{m_j},\qquad h_j(x,w)=\sum_{|\alpha|\geq m_j}A_\alpha(w)x^\alpha. \qquad\text{(PF5)}$$

For a fixed centre point $(0,w_0)$, its degree-$m_j$ homogeneous normal polynomial $P_{w_0}(X)=\sum_{|\alpha|=m_j}A_\alpha(w_0)X^\alpha$ is nonzero. Otherwise (PF5) shows that every term in the Taylor expansion at $(0,w_0)$ has total degree at least $m_j+1$, contradicting the exact point order.

The $x_\ell$-chart of the real projective blow-up has coordinates $(e,v,w)$ and analytic map $x_\ell=e$, $x_i=ev_i$ for $i\ne\ell$, with $w$ unchanged. Substitution in (PF5) proves the analyticity of (PF2). On the fibre over $(0,w_0)$, its restriction is $P_{w_0}(v_1,\ldots,v_{\ell-1},1,v_{\ell+1},\ldots,v_c)$. This is a nonzero polynomial: dehomogenization in any one coordinate is injective on the vector space of homogeneous polynomials of this fixed degree. A nonzero real polynomial is not identically zero on any open neighbourhood of any finite $v$-point. If the germ of $h_{j+1}$ at such a fibre point were divisible by $e$, its restriction to $e=0,w=w_0$ would instead vanish on a neighbourhood. This contradiction proves that the exceptional equation divides none of these germs. The charts cover the whole projective normal fibre. For $c=1$, the dehomogenized polynomial is a nonzero constant, so the argument includes Cartier centres.

Proof for the old divisors and the total exponents. In the same coordinates an old hypersurface containing the centre has equation $x_i=0$. If $i\ne\ell$, its pullback is $ev_i$ and its strict transform in this chart is $v_i=0$. If $i=\ell$, its pullback is $e$ and its strict transform is absent from the chart. An old hypersurface not containing the centre has one of the tangential coordinate equations, say $w_k=0$, and its pullback is the same equation. These descriptions exhibit $F_{j+1}=\{e=0\}$ and all nonempty old strict transforms as distinct coordinate hyperplanes. They prove the updated simple-normal-crossing assertion.

Suppose a weak generator vanished identically on the germ of an old strict hypersurface at some point. Its vanishing would hold on a small open piece of that hypersurface. Points of this piece off $F_{j+1}$ are dense, as is immediate from the coordinate description. At any such point the blow-up is an analytic isomorphism and $e$ is a unit; thus the original $h_j$ would vanish on an open germ of the original $H$. That contradicts the assumed exclusion of every local divisor germ. This proves the assertion for old strict transforms as well.

Substitute the displayed pullbacks of old equations and $h_j\circ\sigma_{j+1}=e^{m_j}h_{j+1}$ into (PF3). Every containing old divisor contributes exactly one additional copy of its exponent to the power of $e$; the transverse ones contribute none. The remaining nonzero factors join the analytic unit. This proves (PF4) and the equality (PF3) by induction from $u_0=1$, $h_0=f$.

These are the actual divisorial multiplicities. Indeed, after removing $z_H^{a_{H,j}}$ from (PF3), restriction of the remaining factors to the germ of $H$ is a product of nonzero analytic germs: the weak generator has that property, the unit does, and every other coordinate equation does. The local ring of the smooth germ $H$ is an integral domain, as follows from convergent power series. The product is nonzero, so one further power of $z_H$ cannot divide the total function. At an intersection of divisors this divisorial order is distinct from the point order, which may be larger. $\square$

Division in (PF2) removes a factor from the weak ideal. Equation (PF4) retains that factor in the total function. Neither a reduced support nor a discarded unit is substituted for the equality of functions (PF3).

Cartier steps and the proper analytic map

Cartier lemma. Blowing up a closed smooth analytic hypersurface is canonically an analytic isomorphism. If its local equation is $x=0$ and $h$ has constant point order $m>0$ along it, then near each point of that hypersurface

$$h=x^m v,\qquad v\text{ an analytic unit}. \qquad\text{(PF6)}$$

The weak ideal becomes the unit ideal near the centre, while the total function retains the exponent $m$.

Proof. The centre-divisibility argument (PF5) gives $h=x^m v$ with $v$ analytic. If $v$ vanished at a centre point, the point order of $h$ would exceed $m$; hence $v$ is nonzero there and is a unit on a smaller neighbourhood.

The normal projective fibre of a hypersurface is $\mathbb{RP}^{0}$, a point. Its sole incidence chart has coordinate map $x=e$, with all other coordinates unchanged. The analytic map from the blow-up to the original chart is therefore the identity in these coordinates. These chart inverses agree on their overlaps because they are inverses of the same blow-up map. They give the asserted global isomorphism, including when the hypersurface has no global defining function. Formula (PF2) leaves the unit $v$; the total-transform identity retains $x^m$. The newly recorded divisor need not be an exceptional locus of a nontrivial modification. $\square$

Properness and endpoint. Every blow-up in (PF1) is a surjective proper analytic map between manifolds of the same dimension. If the supplied tower ends with $I_N=\mathcal O_{M_N}$, then $\pi_N:M_N\to M$ is proper and surjective, and (PF3) is a unit times a coordinate monomial at every point.

Proof. Near a centre of codimension $c\geq1$, the blow-up is the incidence manifold $$\{((x,w),[\lambda]) : x_i\lambda_k=x_k\lambda_i \text{ for all }i,k\} \subset V\times\mathbb{RP}^{c-1}.$$ Its chart $\lambda_\ell\ne0$ has the coordinates $(e,v,w)$ already used, so its dimension equals that of $V$, and its projection is analytic. Off the centre the projective coordinate is uniquely $[x]$, giving an analytic inverse there; over a centre point the fibre is exactly $\mathbb{RP}^{c-1}$. This proves surjectivity, including $c=1$.

These local incidence descriptions also retain the manifold separation and countability conditions. Points with different images are separated by inverse images of disjoint base neighbourhoods; points with the same image lie in one Hausdorff incidence neighbourhood and can be separated there. A countable base-chart cover, with finitely many projective charts over each centre chart, gives second countability.

The incidence equations define a closed subset. For a compact set $Q$ contained in such a coordinate neighbourhood, its full inverse image is a closed subset of the compact space $Q\times\mathbb{RP}^{c-1}$, hence compact. The same assertion holds in an open chart disjoint from the centre, where the map is an isomorphism. Now let $K\subset M_j$ be any compact set. Choose finitely many compact coordinate neighbourhoods $Q_1,\ldots,Q_r$, each contained in one of these charts, whose interiors cover $K$. The inverse image of $K$ is a closed subset of the finite compact union $\bigcup_i\sigma_{j+1}^{-1}(Q_i)$. It is compact. Thus $\sigma_{j+1}$ is proper, and finite composition proves properness of $\pi_N$.

At the endpoint every local weak generator is a unit, so it joins $u_N$ in (PF3). This is the asserted unit-monomial expression. The positive-codimension condition rules out an empty projective fibre; it is automatic for centres contained in the zero set of a nonzero analytic germ. In dimension zero the original function is already a unit on each point, and the identity map suffices. $\square$

At a zero of the total pullback, the preceding unit-normalization lemma gives a sign times a coordinate monomial, preserving its crossing hyperplanes. At least one exponent is then positive, as required by that lemma. Away from the zero set a nonconstant unit need not become a constant under an invertible coordinate change.

The original regular-zero locus is protected

Set

$$Z=\{p\in M:f(p)=0,\ df_p=0\}, \qquad U=M\setminus Z. \qquad\text{(PF7)}$$

The centre hypotheses above do not by themselves make the tower an isomorphism over $U$. For $f(x,y)=x$, the origin is a smooth constant-order-one centre with no old divisors, yet blowing it up introduces a projective line over a point where $df\ne0$. Thus the centre-selection rule must enter the argument.

We use this precise additional condition:

(R) At a point where the weak generator has order one and no member of the current divisor family passes through the point, any chosen centre meeting the point has germ equal to the entire smooth weak-zero hypersurface there.

Centres also avoid the unit-ideal locus by the already imposed condition $C_j\subset V(I_j)$.

The full terminal-component rule has (R), with the following meaning of “full.” A chosen centre germ is a whole local component of the terminal equal-invariant stratum, rather than a smaller smooth submanifold merely contained in that stratum. Here is the entire calculation needed at an order-one point without old divisors. Take $t=h_j$ as an analytic coordinate, using $dh_j\ne0$, and present the weak equation with mark one. Its coefficient hypersurface is $N=\{t=0\}$; its sole coefficient below degree one is $h_j|_N=0$. There are no old-divisor entries. In the order/history notation, the word is therefore $(1,0;\infty)$. After shrinking, every point of $N$ has exactly this word, whereas off $N$ the weak ideal is a unit. Thus its equal-word germ is precisely the single smooth hypersurface germ $N$. A full local component through the point is $N$ itself, which proves (R). This verification assumes that the rule chooses a centre with the stated full-component property; it does not construct a global closed centre or prove compatibility of independently chosen towers.

Protected-complement lemma. Every supplied admissible principal tower satisfying (R) restricts to an analytic isomorphism $\pi_j^{-1}(U)\xrightarrow{\;\pi_j\;}U$ at each stage.

Proof. We prove the assertion together with this local description over $U$: the weak ideal is either a unit, or its generator is a smooth coordinate and the divisor family is empty near that point. Initially this holds because $f\ne0$ gives a unit and $f=0,df\ne0$ gives a coordinate by the analytic inverse function theorem.

Assume the description and the isomorphism through stage $j$. At a point with unit weak ideal, the centre is absent on a neighbourhood, so the next blow-up is an isomorphism there and the ideal remains a unit. At a point with nonunit weak ideal, the generator has order one and there are no old divisors nearby. If the point is outside $C_j$, closedness of $C_j$ supplies a neighbourhood avoiding the centre; the same description persists there. If the point belongs to $C_j$, condition (R) makes that centre locally the full smooth weak-zero hypersurface. The Cartier lemma says that the blow-up is an isomorphism there and makes the weak ideal a unit. A divisor can be recorded in this last case, but the alternative “weak ideal is a unit” places no emptiness requirement on the divisor family.

These alternatives cover every point over $U$, and the local inverses agree wherever their domains overlap because they invert the same map. They therefore give an analytic inverse on the entire inverse image of $U$. This proves the induction and the conclusion for every finite composite. $\square$

Combining the endpoint and protected-complement lemmas gives the desired principal-function conclusion for any admissible tower satisfying (R) and reaching the unit weak ideal. The later compact-fibre termination theorem constructs these towers on neighborhoods of every base point. Their independent-tower comparison supplies analytic isomorphisms over full overlaps. The global assembly then applies the proper-gluing lemma. The total-function factorization and the isomorphism over $U$ pass through those comparison maps because they are local properties over the base. No uniform finite tower on the whole noncompact manifold is required.

Completing the cusp calculation

For $f(x,y)=y^2-x^3$, one has $df=(-3x^2,2y)$, so $Z=\{(0,0)\}$. The first blow-up makes the weak zero curve smooth but leaves it tangent to the first divisor. We keep both projective charts at each subsequent blow-up.

The first two charts, with their analytic maps to the original plane, are

$$\begin{array}{lll} x=u,\ y=uv: & f\circ\pi_1=u^2(v^2-u), & E_1=\{u=0\},\\ x=rs,\ y=s: & f\circ\pi_1=s^2(1-r^3s), & E_1=\{s=0\}. \end{array} \qquad\text{(PF8)}$$

The weak factor in the second chart is nonzero at every point of $s=0$. In the first chart its zero curve is $u=v^2$, tangent to $E_1$ at the unique meeting point $(0,0)$. Blow up that point. The two charts over the first chart of (PF8) are

$$\begin{array}{lll} u=ab,\ v=a: & f\circ\pi_2=a^3b^2(a-b), & E_2=\{a=0\},\ E_1'=\{b=0\},\\ u=c,\ v=cd: & f\circ\pi_2=c^3(cd^2-1), & E_2=\{c=0\}. \end{array} \qquad\text{(PF9)}$$

The order of the weak equation $v^2-u$ at the second centre is one, so (PF4) gives the new total exponent $1+2=3$. The second chart of (PF9) has unit weak factor near each point of its exceptional divisor. In the first chart, the two old/new divisor axes and the weak line $a=b$ meet at the origin. All three are smooth, but three distinct branches at one point of a surface do not form a simple-normal-crossing divisor. Blow up that one point.

The two resulting charts are

$$\begin{array}{lll} a=p,\ b=pq: & f\circ\pi_3=p^6q^2(1-q), & E_3=\{p=0\},\ E_1\text{”}=\{q=0\},\\ b=t,\ a=tw: & f\circ\pi_3=t^6w^3(w-1), & E_3=\{t=0\},\ E_2'=\{w=0\}. \end{array} \qquad\text{(PF10)}$$

Their composite maps to the original plane are, respectively, $(p,q)\mapsto(p^2q,p^3q)$ and $(t,w)\mapsto(t^2w,t^3w^2)$. They also check the total pullback formulas directly. At the third centre the weak equation has order one, and both existing divisors contain the centre. The new total exponent is therefore $1+3+2=6$, exactly as in (PF4).

The remaining weak curve is $q=1$ in the first chart and $w=1$ in the second. On their overlap, $w=1/q$ and $t=pq$, so these represent the same curve and its same meeting point with $E_3$. The curve is transverse to $E_3$. The old strict divisors meet $E_3$ at the other two distinct projective directions: $E_1\text{”}$ at $q=0$ and $E_2'$ at $w=0$. The latter point is outside the $q$-chart. At each final intersection, the two local branch equations have independent differentials; no three branches meet.

This also gives the monomial expression directly in each relevant neighbourhood. Near $q=0$, use $(p,q)$, with unit $1-q$. Near $q=1$, use $(p,q-1)$, with unit $-q^2$, giving $-q^2p^6(q-1)$. At other finite points of $E_3$ in that chart, $q^2(1-q)$ is a unit and only $p^6$ remains. In the other chart near $w=0$, the unit is $w-1$, leaving $t^6w^3$. These neighbourhoods cover the complete third projective fibre.

To check the entire surface, retain the unaffected $(r,s)$-chart from the first blow-up and the unaffected $(c,d)$-chart from the second. Together with the two charts in (PF10), these four charts cover $M_3$. Their maps and complete functions are

For completeness, all six pairwise overlaps can be read from these blow-up coordinates. Domains are stated in the coordinates on the left; the inverse domain is stated in the last column.

Each pair consists of mutually inverse analytic maps on the displayed open sets. Substituting in the preceding table proves agreement of the original coordinates and of the total function. These transitions are composites of the three elementary blow-up transitions already given, so they obey the cocycle identities on triple overlaps. The zero values allowed by the stated domains are essential: for example, the $(r,s)$/$(p,q)$ overlap includes $s=q=0$, whereas the inverse formula $p=y/x$ in original coordinates would not be defined there.

In the first final chart the weak zero locus is disjoint from $s=0$ and smooth: on $1-r^3s=0$ one has $r\ne0$ and $\partial_s(1-r^3s)=-r^3\ne0$. In the second it is disjoint from $c=0$ and smooth: on $cd^2-1=0$ one has $d\ne0$ and $\partial_c(cd^2-1)=d^2\ne0$. In each of the last two charts the weak curve and old exceptional component are disjoint parallel coordinate lines, each transverse to $E_3$. At every point of $M_3$, therefore, at most two branches of the total zero divisor meet and their local defining equations have independent differentials. This proves the simple normal-crossing assertion and all four displayed multiplicities.

All three centres lie above the origin. The properness argument proves that their composite $\pi_3:M_3\to\mathbb R^2$ is proper and surjective; it is an analytic isomorphism over $\mathbb R^2\setminus\{0\}$. The unit-normalization lemma gives sign-only monomial coordinates at every point above the origin. Thus this example proves the complete function-resolution conclusion, with all multiplicities and every projective direction retained.

One may also continue to the unit weak ideal. The final weak-zero curve is a closed smooth analytic hypersurface, and it meets the existing divisors normally. Blowing it up is a Cartier identity step with weak order one. It records the residual-curve exponent and makes the weak ideal a unit everywhere, without changing the manifold or the total pullback formulas.

Open the full-size diagram · Reproduce the diagram.

Exercise: distinguish the weak order from the total order

At each of the three point centres, compute the weak order and the order of the complete function. Recover the new exceptional multiplicity, and explain why two point blow-ups do not suffice for simple normal crossings.

Solution. Initially the weak and total functions coincide, and $y^2-x^3$ has order two. Thus $E_1$ has multiplicity two. At the second centre $(u,v)=(0,0)$, the weak factor $v^2-u$ has order one, while $u^2(v^2-u)$ has order three. Its new exceptional exponent is $1+2=3$. At the third centre $(a,b)=(0,0)$, the weak factor $a-b$ again has order one, while $a^3b^2(a-b)$ has order six. Both old exceptional divisors contain that centre, so its new exponent is $1+3+2=6$. After two blow-ups there are three distinct divisor branches at one point of a surface. An analytic coordinate change has an invertible derivative and preserves distinct tangent lines; it cannot turn those three branches into the at most two coordinate hyperplanes available in dimension two. The third point blow-up is therefore needed in this tower.

Exercise: identify the final curve and normalize the three crossings

Write $C_3$ for the final strict curve. Show that the whole curve is $q=1$ in the $(p,q)$-chart and that $\pi$ restricts to $p\mapsto(p^2,p^3)$. Find exact sign-monomial coordinates at the three points where $E_3$ meets $E_1\text{”}$, $E_2'$, and $C_3$.

Solution. In the $(r,s)$-chart a weak zero satisfies $r^3s=1$, hence lies in the overlap $r\ne0$ and has $p=1/r, q=r^3s=1$. In the $(c,d)$-chart it satisfies $cd^2=1$, hence lies in the overlap $cd\ne0$ and again has $q=1$. In the $(t,w)$-chart it has $w=1$, hence lies in the $(p,q)$-chart with $p=t,q=1$. Thus the entire curve is covered by $q=1$, with $p\in\mathbb R$; the direct map in the table gives $(x,y)=(p^2,p^3)$. This is a computation on the resolved curve, not an assertion that the singular cusp is an analytic manifold.

Near $E_3\cap E_1\text{”}$, namely $(p,q)=(0,0)$, set $P=p(1-q)^{1/6}$, $Q=q$. The positive real sixth root is analytic near $q=0$, and the coordinate change has nonzero Jacobian. Then $f\circ\pi=P^6Q^2$. Near $E_3\cap E_2'$, namely $(t,w)=(0,0)$, set $T=t(1-w)^{1/6}$, $W=w$; this gives $f\circ\pi=-T^6W^3$. Finally, near $E_3\cap C_3$, write $z=q-1$ and set $P=p(1+z)^{1/3}$, $Z=z$. Since $f\circ\pi=-p^6(1+z)^2z$, this gives $f\circ\pi=-P^6Z$. All roots are taken only of positive units near the specified points, and each change preserves the displayed coordinate crossing hypersurfaces. No root is taken of the possibly zero exceptional coordinate.