smooth points of p -adic subanalytic sets

manuscripta math. 80, 45 - 71 (1993) manuscripta mathematica

Spfinger-~rlag 1993

Smooth points of p-adic subanalytic sets

Zachary Robinson

A p-adic subanalytic set shares with a real subanalytic set the fundamental property that its singular locus is itself subanalytic. Furthermore, given a p- adic subanalytic function f with domain contained in 7]~, there is an integer L such that for any point z0 6 ~7~ in a neighborhood of which f is defined, f has a Taylor approximation up to order L at x0 if, and only if, f is analytic around x0. These results extend to the p-adic fields real variables theorems by M. Tamm [21].

I. Introduction

J. Denef and L. van den Dries [6] initiated the study of subanalytic sets over Qp. The theory of real subanalytic sets was first developed by A. Gabrielov [9], R.. Hardt [11] and H. ttironaka [12], building upon the work of S. Lojasiewicz [15] on real semianalytic sets. Although the theory of p-adic subanalytic sets closely parallels that of the real-variables case, the p-adic techniques differ by an essential use of quantifier elimination.

The aim of this paper is to show that a p-adic subanalytic set (Definition 5.1 and Corollary 5.3) shares with a real subanalytic set the fundamental property that its singular locus is itself subanalytic (Theorem 5.9). We also show, given a p-adic subanalytic function f (Definition 5.1) with domain contained in 27~ n,

m that there is an integer L such that for any point z0 6 7]p in a neighborhood of which f is defined, f is strongly differentiable (Definition 5.6 and remarks following) up to order L at z0 if, and only if, f is analytic around z0 (Theo- rem 5.8).

These results extend to the p-adic fields real variables theorems by M. Tamm [21]. As does Tamm, we use desingularization to reduce the problem of finding the singular locus to the problem of finding the graphic points (Definition 3.1) of an analytic map. Tamm's proofs, however, rely heavily on properties of mero- morphic functions over ~3. (The reader is also referred to the proof of Tamm's Singular Locus Theorem in the exposition of the theory of real subanalytic sets by E. nierstone and P. Milman [3]; in particular to [3], Lemma 7.8.) Many of these properties do not have suitable ultrametric analogs. Our approach instead

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is to use data obtained from the rational coefficients of an explicit parameterization of the solutions g to the graphic points functional equation of Definition 3.1. Lemma 4.5 links divisibility properties of these coefficients to the singular be- havior of subanalytic sets and to the Uniform Krull Intersection Theorem of Theorem 4.4.

In Section 3, we establish the required properties of graphic points of an analytic map. The idea of graphic points is due to B. Malgrange [16]. A graphic point of an analytic map is, roughly speaking, a point of the domain which has a neighborhood whose image is contained in the graph of an analytic function (whence the relation to smoothness.) We first obtain a parameterized solution to the graphic points functional equation. Together with the Krull Intersec- tion Theorem, this yields the main result of Section 3: the set of non-graphic points of an analytic map given by strictly convergent power series over a field of characteristic 0, complete in a non-trivial ultrametric valuation, is actually the zeroset of strictly convergent power series (Theorem 3.7). This extends to ultrametric fields results of Maigrange over r

In Section 4, we give a definition of approximately graphic (Definition 4.1). Following Tamm, we prove stationarity for such approximations (Lemma 4.3). This is used to obtain the above differentiability property of p-adic subanalytic functions. Using the Uniform Artin-Rees Theorem of A. Duncan and L. O' Car- roll [7], we next prove in Theorem 4.4 a uniform version of the Krull Intersection Theorem for excellent rings. In the special case that the given ring is an affinoid algebra (affinoid algebras are excellent), we give a different proof of The- orem 4.4 using the method of Lemma 4.3. Theorem 4.4 allows us to establish in Lemma 4.6 an estimate on the bound shown to exist by Lemma 4.3. This estimate is stated in terms of divisibility properties of the coefficients of the solution G to the graphic points equation given in Lemma 3.2; moreover, the coefficients of G are given recursively. Theorem 4.7 collects the results of this section into an Approximation Theorem for the solutions of the graphic points functional equation.

In Section 5, finally, we prove the p-adic analogs of Tamm's theorems con- cerning smooth points and differentiability. We first extend the Uniformization Theorem of Denef and van den Dries to get a desingularization theorem (Theo- rem 5.2) analogous to Hironaka's Uniformization Theorem for real subanalytic sets. This enables us then to apply the results of Sections 3 and 4. We also use the desingularization of Theorem 5.2 to show in Corollary 5.3 that the p-adic statement analogous to the definition of a real subanalytic set given by ttironaka in [12], Definition 5.1 yields the same class of sets as the definition of a p-adic subanalytic set given by Denef and van den Dries in [6].

L. Lipshitz [13], Section 4 and [14], and H. Schoutens [19], Chapter III, using quantifier elimination techniques, have introduced theories of rigid subanalytic sets; that is, subanalytic sets defined over various power series rings with coefficients in an algebraically closed field of any characteristic, complete in a non-trivial ultrametric valuation. Their quantifier elimination techniques apparently cannot be applied over the corresponding Tate rings of power series with polyradius of convergence > 1 (the rings of strictly convergent power

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series.) Schoutens's theory is based on the smaller rings of power series with polyradius of convergence > 1 (the rings of overconvergent power series,) and Lipshitz's theory is based on larger power series rings (the reader is also referred to W. Bartenwerfer [1], where Lipshitz's rings are used to establish boundedness properties of analytic maps.) As in the p-adic case, many of the fundamental properties of real subanalytic sets are shown to hold as well for rigid subanalytic sets. The rigid analogs of Tamm's theorems, however, are still open. Owing to difficulties which stem from the lack of compactness in the algebraically closed case and from the nature of the rings of power series that are used, transposition of the present arguments to the rigid case would not be a routine matter.

The author thanks Craig Huneke for pointing out that Theorem 4.4 follows from the Uniform Artin-Rees Theorem of Duncan and O'Carroll. I also thank Leonard Lipshitz for several valuable discussions during the preparation of this paper.

2. N o t a t i o n a n d c o n v e n t i o n s

Throughout this paper, K denotes a field of characteristic 0, complete in the non-trivial ultrametric valuation [ .[ :K --* IR. For n G IN, we give K" the K- module norm [ (z t , . . . , z ,~) l := maxt<i<n Iz l. By R, we denote the valuation ring of K.

Let X = ( X 1 , . . . , Xm) be an m-tuple of variables. Let/3 E IN" be a multi-

index and let ej E IN" denote the multi-index with j t h coordinate 1 and 0 elsewhere. Put 181 := ~1 + . . . + ~,~, where/3 = (~1 , . . . , tim). A power series f = ~_,0 aoX # is called strictly convenjent iff limlal...oo [aa[ = 0, and in this c e, the Gauss norm Ilfll o f f is defined to be maxt~ laal. A strictly convergent power series f determines a function f : R m ---* K.

A power series f is called convergent iff, upon replacing the variables X by points z = ( z l , . . . , z,n) in a sufficiently small neighborhood of 0 in R m, the result is a series convergent in K.

We use the following notation to denote various power series rings over K:

K(X1,..., Xrn) denotes the ring of strictly convergent power series,

with Gauss norm I1"11,

K{X1,..., Xm} denotes the ring of convergent power series,

K[[X1,..., X,~]] denotes the ring of formal power series, and

K ((X1,..., Xm)) denotes the field of fractions of KITX].

The Gauss norm is an ultrametric valuation on K(X) which coincides with the sup norm in the sense that If(z)[ < [Ifll for any z E R " , and equality holds for some z, [z[ < 1, with coordinates in the algebraic closure of K. The rings If(X), K{X} and K[[X]] are each Noetherian UFDs. (The reader may refer to [4] or to [8].)

Consider the Taylor expansion map

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Tx: K[[X~ ~ K[~X~[[YII: f ~ f ( X + Y).

Since Txg(O) = g(X), if g ~ 0 then Txg is a unit in K((X))[[Y]]. Thus Tx extends to the homomorphism

Tx : K ((X)) --~ K ((X)) [IY~: [" ~ Tx f g Txg

which satisfies the identities Tx f(O) = f,

and by the Chain Rule,

~ (-) cgy~ Tx.f = Tx - ~

for any f E K ((X)). We will also make use of the K-linear map

~x: K((X)) --. K((X)) [[Y~: f ~ ( T x f - f).

Note that ~x f (0 ) - 0. We regard Tx f as a power series in the variables Y with coefficients param-

eterized by X. We may make certain substitutions both for X and for Y. For example, given g = g(Z) E K[[ZI, . . . , Zn]], ~l,. . . ,~an E K(X) and Zo E R m,

the expression (T~o(x+ro)-~o(xo)g) o T~(+ro~o denotes the power series

g ( ~ ( x + xo) - ~ (x0) + ~ ( x + ~o + Y) - ~ ( x + ~0)) =

= g ( ~ ( x + ~0 + Y) - ~(~0)) e K[[xHY~.

The symbol D denotes the Jacobian matrix of partial derivatives with respect to the given variables.

Finally, the next condition will be used throughout this paper.

Def in i t ion 2.1. Let ~o : - ( ~ l , . . . ,~'n) and f := ( f l , . . . , / r ) , where ~i , f j E K[IXt,. . . ,Xm]]. Then (~0, f ) is said to satisfy the r a n k c o n d i t i o n iff some (n x n) minor of D~o is not identically zero in K[IX]] and every (n + 1) x (n + 1) minor of D(~o, f) vanishes identically.

3. Graphic po in t s

The aim in this section is to develop an ultrametric analog (in the strictly convergent power series ring over a field of characteristic 0) of the real and complex variables theory of graphic points due to Malgrange [16].

Def in i t ion 3.1. Let (~, f ) : R m -* K n x K be given by strictly convergent power series. Suppose (~, f ) satisfies the rank condition. A point x0 E R m is said to

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be a g r a ph i c po in t of (~o, f ) iff there is a power series g E K{Z1 , . . . , Zn} such that

+ = g( (x + -

whenever x is sufficiently near 0 in R m.

This terminology is explained by noticing that z0 is a graphic point of (~, f ) exactly when there is a small neighborhood U of x0 in R m such that (~o, f)(U) is contained in the graph of a function analytic at ~(x0). Theorem 3.7 states that the set E of non-graphic points of (~o, f ) forms a zeroset in R ~.

Here is the strategy of this section. Using Taylor expansions and the formal Inverse Function Theorem of [5], Proposition IV.4.7.10, we are easily able to demonstrate in Lemma 3.2 the existence of a parameterization G = G(X, Z) of the solutions of the graphic points functional equation of Definition 3.1. Lemma 3.3 produces a recursion formula for the coefficients of the unique formal power series G( Z) E K (( X)) [[Z]] which satisfies

T x f = G o T ~

Moreover, by Corollary 3.4, when q0 and f are strictly convergent, the coefficients of G are ratios of elements of K{X), where the denominators can be taken to be powers of an (n x n) minor of Dqo. The problem of finding graphic points of (qo, f ) is reduced in the proof of Theorem 3.7 to finding points of R "~ about which the numerators of the coefficients of G are formally divisible by the denominators. From the convergence result of Lemma 3.6 it follows that such points are precisely the graphic points. To prove that the set E of non-graphic points is actually a zeroset, we use the fact that the strictly convergent power series rings are UFDs and note that the Krull Intersection Theorem implies, along each irreducible component of the zeroset of the denominators, that formal and strictly convergent divisibility are the same (Lemma 3.5).

Because we want to apply them in a somewhat different setting in Sec- tion 4, Lemma 3.3 and Corollary 3.4 are stated in slightly greater generality than needed in this section.

Lemana 3.2. Let X = (X1 , . . . ,Xm) , Y = (I11,...,Ym) and Z = (Z1 , . . . , Z , ) be variables, m > n > 1. Let ~ol,..., ~on, f e K~X]], where (~o, f ) satisfies the rank condition. Then there exists precisely one G E K((X)) [[Z]] such that

Tx f ( Y ) = G o T~ ~o(Y).

Proof. To prove the existence (and uniqueness) of G, we use the formal Inverse Function Theorem, working over g ((Z)) [[Y]]. Since some (n x n) minor of Dqo

is not 0, by reordering the Xi, we may assume that det ( ~ ) l < , , j < n ~ O. Put

~(Yl,...,Ym) := (T~l,--.,T~n,Yn+1,...,Ym).

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Since det ( ~ . ) ~ 0, we must have det D#ly= 0 ~ O. By the Inverse Function

Theorem, there is a unique ft E K ((X))llY]] m with ft(0) = 0 such that

�9 o f t = f t o ~ = ( Y 1 , . . . , Y r n ) .

In particular, T~ta o ft(Y) = (Y1, . . . , Yn). By the Chain Rule, D [ ( T ~ ~o, T x f ) o ft] = [D(T~ 9, T x f ) ] o f t . Dft. Since

det Dft ~ 0 and since rank[D(T~o, T x f ) ] o f t = rankn(~o , f ) = n, we must have rank n [(T~ T x f ) oft] = n. Therefore,

0 0 1OWn+ ~ ( T x f o f t ) , . . . , OYm ( T x f o f t ) ~ O.

This implies that

T x f o f t(Yl, . . . , Urn) ~ T x f o ft(Yx, . . . , Yn, O, . . . , O),

since K ( ( X ) ) has characteristic 0. Finally, put

G(ZI . . . . ,Zn) := T x f o f t ( Z l , . . . , Z n , O , . . . , O ) E K((X))[IZ]].

We have T x f o f t = G o ( T ~ of t ) .

Composing with f t - t yields

Txf = Go T~ ~,

which is what was to be shown. We now obtain a recursion formula for the coefficients of G.

[]

L e m m a 3.3. Let fib, . . . , # n , H E K((X))~Y]] and G E K((X))[IZ]]. Put ~ := (@1,. . . , @n). Suppose:

O) ~(o) = o, (ii) some (n • . ) minor of ( ~ , ) is .o , the zero power ser~es,

(ii i) ~ i = T x ~ Y = O '

c,v)o. (~ ~I;,7 y : o , ) l < i < m , and

(v ) G o O = H.

Write G = ~_~.,.Z '8 and H = z_..,V'h"Yaa! . Then fo = ho and # a

Of#.. _ E ft~+e i 1 < i < m, It~l > 1. (3.3.1) OXi OWl [Y--O] ' -- j = l

If, in addition, H = Txho, lhen (3.3.1) holds when/3 = O.

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Proof. Putting Y = 0 in (v) yields f0 = h0. We show by induction on/3 that the f , satisfy condition (3.3.1) and in addition satisfy

OI~IG T x f p = ~ o ~ (3.3.2)

when ]/?1 > 1. Indeed, using the Chain Rule to differentiate (3.3.2) yields

= (ot,+~Jta ) 0~, OOyiTxf~ ~-~ \ ~ o~ -~i l < i < m . (3.3.3) j=l

When Y -- 0 we obtain

OXi = f ,+ej - ~ i , l < i < m, j = l Y=O

so that (3.3.1) follows from (3.3.2) for any ft. On the other hand, applying T x to (3.3.1) and using (iii) yields

OOyi T x f , ,r Of , n ) ~ ---- l x - ~ i = Y ~ ( T x f , + e j , 1 < i < m. j = l

By (ii) and Cramer's Rule, (3.3.3) and the above equation immediately imply

T x f ' + ~ i = OZa+eJ o~ , l < j < n ,

so that (3.3.2), thus also (3.3.1) holds for/3 + ej, 1 < j < n. Since (3.3.1) follows from (3.3.2), it remains to show that (3.3.2) holds for

I~1 = 1. Using the Chain Rule to differentiate (v) yields

aYi = o ~ - ~ / , l < i < m . (3.3.4) j = l

When Y = 0 we have

0Y/ Y=O j=l "~i ly=o] ' 1 < i < m. (3.3.5)

Applying Tx to (3.3.5) and using (iii) and (iv) yields

- = ~ '~ ' (Txf , i 1 < i < m. O5 j = l

By Cramer's Rule, (3.3.4) and the above imply that (3.3.2), thus also (3.3.1) holds for I~[ = 1.

When H = Txho, since f0 = h0, we have H = Tx fo . When fl = 0, note, in this case, that (3.3.2) for ~ = 0 reduces to (v). Thus, when H = Txho, (3.3.1) also holds for fl = 0. []

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We now turn to the strictly convergent case, using the recursion formula (3.3.1) to obtain more information about the coefficients fa.

C o r o l l a r y 3.4. Let ~ ,G and H satisfy conditions (i) - (v) of Lemma 3.3, where in addition r 1 6 2 H E (K(X))~ /~ . Let 6 be a nonzero (n x n)

minor of ~ Y=O

tha* b

Proof. When fl = 0, take ga to be h0. When I/~l > o, we will use equations (3.3.1) and (3.3.5). For convenience, let us assume that the Y/ have been

ordered so that 6 = det a o ) r 0. Let the adjoint matrix of (~r Y=0 l<_i,~__.,

' 1 ) be denoted by ( ~ " y=o) adj i I Y = O / l ~ i , j < n

Solving equation (3.3.5) for the fei, we obtain

1 ( O#/ ,~ adj ( f e a ' " " f * ' ) t = ~ \ ~ i l y = o ] (he"""he ' ) t "

ge . Thus, for some g,j E K{X), fej = ~ , 1 <_ j <_ n. We have completed the

case where ]1~1-< 1. Now induct on/~. Assuming f~ = 3~7~1, g# E K(X) , Ifl] > 1, we have

of a 6 exeT, - 21 1 o, _ g# FXT/ l < i < m . ( .)

cOXi -- 621/~1 +l '

Solving equation (3.3.1) for the fa+,~, 1 < j < n yields

By (*),

= r = 0 ]

f#+% = gJ+e~ di21#l+~

for some ga+e~ E K(X) , 1 < i < n. [] Next we use the Krull Intersection Theorem to show that formal and strictly

convergent divisibility by an irreducible element of K (X) coincide at any point where the denominator vanishes. This is useful in the proofs of Lemma 3.6 and Theorem 3.7.

L e m m a 3.5. Let xo E R m and suppose f ,g E K(X) with g irreducible and g(x0) = O. I f for some h E K[[X]], f ( X + xo) = he(X + xo) then h E K<X).

Proof. Since g is irreducible, so is g(X + x0). Hence the ideal g(X + x0). K(X) is a prime ideal. (Recall that K(X) is a UFD.) Let m be the maximal ideal of

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K(X) generated by the variables. Since g(zo) = O, g(X + xo) . K(X) C m. By the Krull Intersection Theorem [17], Theorem 8.10,

g(X + Xo). K(X) = N (g(X + xo). K(X) +mr). (.) lEIN

Now, f ( X + xo) = hg(X + z0) for some h E K[[X]l if, and only if, f ( X + z0) E g(X + xo). K(X) + m l for every ~ E IN. By (*), this happens if, and only if, f ( X + xo) E g(X + xo) . K(X) . Since K~X]] is an integral domain, the lemma follows. El

The following lemma is analogous to a convergence result in [16] (or to the much more general result in [10].)

L e r n m a 3.6. Let ~ := (~P l , . . . , ~n ) , where tat E K { X I , . . . , X m } . Suppose that ~(0) = 0 and that some (n • n) minor of D~ is not identically zero. Let g e KEZ1,. . . , Zn]] and suppose g o ~ e K{X} . Then g E K{Z} .

Proof. Put 6 := det (~176 and f := g o ~o. Since the generic rank of

is n, we may reorder the Xi so tlqat 6 ~ 0. Since the characteristic of K is 0, if 6(0) # 0, then g e K { Z } by the analytic Inverse Function Theorem (see [20], LG p.2.10), arguing as in Lemma 3.2.

It remains to treat the case when 6(0) = 0. Replacing X by 6 �9 X for some E # 0 with ]e] sufficiently small, we may assume that ~1, . . . , ~n, f , 6 E K(X) .

By Lemma 3.2, there is a unique G = ~'~ {fT,.Z • E K((X)) [[Z]] with

T x f = a o

Furthermore, the f# satisfy condition (3.3.1) and, by Corollary 3.4, for each f? there is a ga E K(X) such that fa = T~-~'" Expanding both sides of the equation f = g o ~ yields

Tx f = T~(x)g o 7~xlO. This equation has a unique solution, so

T r = G(Z),

whence each fp E K~X]].

Cla im. There is some e # 0 such that fp(e. X) is strictly convergent for every /3 E IN".

Let 61, . . . , 6r be the prime factors of 6 which vanish at 0 and let 6 r+ l , . . . , 68 be any other prime factors of 6. Take e r 0 with le] sufficiently small so that 6i(e. X) is a unit in K(X) , r + 1 < i < s.

We have P g~ P

fz = ~ = q = 6 ~ ' . . . 6 ~ 6 " '+ ' '~'' r + l " ' " 68

where cq,...,(x, E IN and where p and q have no prime factor in common. Suppose for some i, i < i < r, that m > 0. Since I~ E gEXll, {T, E KIIX]].

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Since 6,(0) = 0, by Lemma 3.5, ~ e K(X) , contradicting our assumption that p and q have no prime factor in common. Thus ai --" 0, 1 < i < r, and fp(6 . X) e K(X) . This proves the claim.

Replacing X by e �9 X, we may assume now that each fOE K(X) . Consider the power series G = ~ Z ~ e K(X)[[Z]]. Note that g =

~ ~ Z ~ because Tr = G(Z) and ~o(0) = 0. To show that g con- verges, we make an estimate on the Gauss norms of the f~.

C la im. There is some fixed M E IR such that for all/3 e IN", llf~li S MI~I IIf01l

. Pick an entry h of the Let be the adjoint matrix of l<i,j<n

adjoint with maximum Gauss norm (so h ~ 0) and put

Ilhll M := [[6[["

Since M ~ 0, there is nothing to show when/3 = 0. Induct on/3. By (3.3.1) we have

Ofo = ~ fo+e~ q ~~ , 1 < i < n . OXi OXi j=l

Solving for the fo+~j, we obtain

/' O~oj ~ adj /, Ofz Ofp ) t 6. ( f Z + , , , . . . , b + , . ) t = \~-~/ \b--~'"" b-x-.) "

Since the Gauss noim is an ultrametric valuation on K(X) , we have

< M [[f~[[

<: MIZ+eJl H/0ll,

as desired. This proves the claim. Since the Gauss norm and the sup norm coincide on K(X) , from the claim

we have for each lf#(o)l _< Ilf/~ll-< Mlzl llfoll.

It follows immediately that g = ~'~. ~ . Z ~ e K{Z} . []

T h e o r e m 3.7. Suppose that (~o, f ) : R m --* K n • K is given by strictly convergent power series and that (~o,f) satisfies the rank condition. Let 6 be a nonzero (n x n) minor of D~o. Then the set E C R m of non-graphic points of (~o, f ) is a union of some of the irreducible components of V(6) in R m.

Proof. By Lemma 3.2 and Corollary 3.4, we have

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TxS = Oo ~x~, where

and g/~ E K(X).

= X-" g~ z ~ o(zl,...,z,) ~

C l a i m . x0 E / ~ is a graphic point of (~, f ) if, and only if, for every ]~ E IN'~

gp(X + Xo) e K[[X1]. ~(X + x0)~l~l

(==~) Suppose x0 E R m is a graphic point of (~ , f ) . Then there is some g E K{Z} such that

T:~of(y ) = g o T~

for all y sufficiently near the origin in R m. Hence for all x and y sufficiently near 0,

Tx+xof(Y) = (T~(x+~o)-~(xo)g) ~ T~

But by Lemma 3.2, this equation has a unique solution, so

T'p(x+'o)-~(~:~ g(Z) = Z g/3(X + xo) Z[ 3 /~ ~!6(X + Xo)21/~l "

This shows that g~(X + Xo) K{X}.

~(X + x0)~lal e

Putt ing X = 0 in the previous two equations yields

1 ( g . ( X + x o ) ) g(z) = ~ \ ~ ( x + ~o)2tpl x=o

Z ~. (3.7.1)

We state (3.7.1) for later use. (~=) Now suppose for every ~ E IN n that

Put

Then

go(X + xo) ~(X + z0)21~l

E K~X]].

~ I ( g.(X +~o) ) g(z):= ~ \ ~ ( x + ~ 0 ) ~ l x=0

Z ~ .

Txof(Y) = g o T~

By Lemma 3.6, g must converge. Therefore, x0 is a graphic point of (~o, f ) . This proves the claim.

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E# := {z0 6 R 'n I ,(X+xop,mg'(x+x~ ~ K[[X]]}. Then by the previous claim, Put

E = UE~. Since there are only a finite number of irreducible components of

V(~) in R " , once we show that each Ea is a union of some of these irreducible components, we are done.

Since K(X) is a UFD, we may write

g# P P - q - ~ , . . . 6 ~ - '

where p and q have no prime factor in common, 6x , . . . , 6r are among the prime factors of 6 and 0 < ~ I , . . . , ~r 6 IN. We will show that

Ep = V(~I) U . . . u V(~r). (3.7.2)

Clearly, Ea C V(61) U . - - U V(Sr). To show the reverse inclusion, take z0 6 V($i), for some i, 1 < i < r. If gp(X+zo) 6 K[IX]] then ~ 6 K[[X]]. By 6(X+zo)~l at Lemma 3.5, then, ~ E K ( X I, contradicting the assumption that p and q have no prime factor in common. Thus z0 E E#, proving (3.7.2) and the theorem. []

It will be useful in the proof of Lemma 5.5 to collect here some consequences of the proof of Theorem 3.7.

Assume the hypotheses of Theorem 3.7. By Corollary 3.4, there are elements g~ E K(X) such that

Txf = Go T~ ~,

where G = ~ g-----~-~ Z ~.

Find p#, q# 6 K ( X I such that p# and q# have no prime factor in common and such that

gp P~ 621# I - - q �9

The irreducible factors of q# are among the irreducible factors of 6. By equation (3.7.2), z0 ~ E if, and only if, q~(zo) ~ 0 for every fl 6 IN n. On

the other hand, by definition, x0 ~ E if, and only if, for some g 6 K{Z} ,

f (x + ~0) = g(~(x + ~0) - ~(~0))

whenever x is sufficiently near 0 in R m. By equation (3.7.1), for such an z0, we have

g(z)= Z p~(~0) z ~ .

This proves the following.

C o r o l l a r y 3.8. (Under the assumptions of Theorem 3.7.) There are elements p~, q~ 6 K (X) which share no prime factors such that Tx f = G o T ~ , where G = ~ ,~.gZ a and where the factors ofq a are among the factors of 6. Let E

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be the set of non-graphic points of (~o, f ) . Then the the following conditions are satisfied.

E = UV(q#) (3.8.1)

Furthermore, suppose zo ~ E is a graphic point of(~, f ) ; i.e., suppose for some g E K{Z} that

f ( x + x0) = + -

for z sufficiently near 0 in R m. Then

= V " pz( o) g(z) (3.8.2)

D

4. A p p r o x i m a t i o n

The two goals of this section are (i) to show that there is a uniform way to decide whether the graphic points equation of Definition 3.1 has a solution by searching for approximate solutions "up to order L," and (ii) to estimate the size of the uniform bound L.

Lemmas 4.3 and 4.6 each show why a uniform bound L exists; in particular, although it depends on Theorem 3.7, it does not follow "automatically" from the results of Section 3. Uniformly reducing the problem of finding the infinitely many coefficients of a solution of the graphic points equation to the problem of finding the finitely many coefficients of an approximate solution allows us, in Section 5, to apply the analytic elimination theory of Denef-van den Dries to obtain our main results on smooth points and differentiability. It is worthwhile to note that Theorem 5.9 on smooth points can be proved without recourse to the results of this section: one shows that the set of non-s-fold fiberwise graphic points (of a collection of maps as in Definition 5.4) is a finite union of images under analytic maps of zerosets in R 2m. Theorem 5.8 on differentiability, however, cannot be proved without the results of this section.

Lemma 4.3 asserts the existence of the desired uniform bound. Using the Uniform Artin-Rees Theorem of Duncan-O'Carroll, we estimate its size in Lemma 4.6. We collect these results as an Approximation Theorem for the solutions of the graphic points functional equation (Theorem 4.7).

Defini t ion 4.1. Let 91, . - . , ~,,, f e K(X), where (~, f ) satisfies the rank condition. Let l E IN. A point zo E R m is said to be approx imate ly graphic up to order l with respect to (9, f ) iff there is some g ~ K [ Z I , . . . , Z,] such that

T~:of(X) - g o T~o~(X ) mod m t,

where m is the maximal ideal of K ( X ) generated by X1, . . . , Xm.

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Obviously, if z0 is graphic then it is approximately graphic up to order g for every L

The proof of Lemma 4.3 involves an application of the Baire Category The- orem, for which we need a geometric definition of dimension of a zeroset. Defi- nitions of K-analytic map and K-analytic manifold are given in [20], LG Chap- ter 3; in particular, a K-analytic manifold has a well-defined dimension.

De f in i t i on 4.2. Let S be any subset of R m. By d i m s we denote the greatest integer d for which there is a dimension d K-analytic submanifold M of R m with M C S.

In this section, we are interested in the case that S is a zeroset in R m. In Section 5, we are interested in subanalytic subsets S C 77~, in which case this is precisely the definition of dimension given in [6], w When S is an irreducible zeroset, d i m s coincides with the Krull dimension of K ( X ) / I ( S ) ; indeed, dim Sing S < dim S and Reg S is a K-analytic manifold of dimension dim S (see [8], Proposition II.7.3.)

L e m m a 4.3. Suppose the map (~o,f) is as in Definition 4.1. There is an integer L E IN such that for every point z0 E R m, i fzo is approximately graphic up to order L then xo is a graphic point of (~o, f ) .

Proof. For each e E IN, let Et denote the set of points in R " which are not approximately graphic up to order g and let E denote the set of non-graphic points of (~, f ) . Clearly,

0 = E o C ..- C E~ C E~+~ C - . . C E.

In this notation, the theorem states for some L that EL = E. For any given z E R m, the equation T ~ f ( Z ) =__ g o T~ mod m t is a

system of finitely many linear equations in the Taylor coefficients of g up to at most order L The condition that this system should have a solution is the same as requiring the ranks of the augmented matr ix and the matr ix of system coefficients to be equal. These matrices have entries given by strictly convergent power series in z. Therefore, each Et is a union of differences of zerosets, each given by strictly convergent power series.

C la im. Suppose E C R m is a zeroset given by strictly convergent power series. Suppose for each s E IN that El is a finite union of differences of zerosets of

L strictly convergent power series. If Ut__.0Et = E then for some L E IN, tUoEl= =

E. Induct on dim E. If dim E -- 0 then E is finite and the claim clearly holds. Suppose d i m E > 0. Since K I X I is Noetherian, E is the union of finitely

many irreducible zerosets. We may as well assume, then, that E itself is irreducible. Now, an irreducible zeroset V has the following useful property. If W is a

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zeroset, W C V and dim W = dim V, then W -- V. By the Baire Category The- orem, for some t l e IN, dim Et, = d imE. Write Et, = (VI\W1)U...U(Vr \Wr) , where each Vi and Wi is a zeroset. By splitting into irreducible components, we may assume that 1~ is irreducible, Vi D Wi and dim ~ > dim Wi, 1 < i < r. We have for some i that d i m ~ = d i m E tl = d imE, thus, since E is irreducible, Vi = E and Wi D E \ Etl. Since dim Wi < dim E, we may apply the induction hypothesis to Wi = Ut>oWi MEt, obtaining an ~ E IN such that

t L Wi = t-~-o Wi MEt. Let L _> tt,g2; then E = t=oU Et. This proves the claim.

By Theorem 3.7, the theorem follows from the above claim once we know that E = Ut>oEt.

We have Ut>0Et C E. To see that Ut>oEt = E, notice that z0 ~ Ut>oEt implies that each finite subsystem of the infinite system of linear equations given by

T~ o f = g o ~ o ~' (*)

has a solution. Hence the entire system (.) has a solution; i.e., for some g E K[[Z~, T=of = g o T~o~. (A model-theoretic proof follows. Let f be any field, A = ( a i j ) i , j e ~ E F ~• and b = (bl)ie~ E F ~ . Suppose for each fixed i that all but finitely many of the aij are 0 and suppose each subsystem of Ax = b consisting of the first finitely many rows has a solution over F. Passing to an R0-saturated field extension L D F, we find a ~ E L r~ for which A~ = b. Extend {1) to an F-vector space basis of L and let r : L ~ F be the projection on the subspace generated by 1. Then r and A commute. Hence x := ~r(~) E F ~ satisfies Ax = b.) By Lemma 3.6, this implies that z0 ~ E. 13

The Krull Intersection Theorem was crucial to the proofs of Lemma 3.6 and Theorem 3.7. In the remainder of this section we will discuss a uniform statement of the Krull Intersection Theorem (Theorem 4.4) for the rings K (X), and the relation of the bound established in Lemma 4.3 to divisibility properties of the coefficients of the power series G E K((X))[[Z~ given by Lemma 3.2.

A K-algebra A is called K.affinoid iff A is isomorphic to a quotient algebra K ( X ) / J for some ideal J of K(X) . (See [4], Definition 6.1.1.1 and Proposi- tion 6.1.1.3.) In particular, K(X) is K-affinoid. Any K-affinoid algebra is an excellent ring [2], Satz 3.3.3.

T h e o r e m 4.4. Let A be an excellent ring, p a prime ideal of A and h E A \ p. Then there is a fixed integer L such that for every proper ideal I D P, h L

+ I').

Proof. Put R := A~ p and let h ~ 0 be the residue of h in R. It suffices to show that there is an L E IN such that h ~ m L for all maximal ideals m of R. Since R is a quotient of an excellent ring, it is an excellent ring. Thus by the Uniform Artin-Rees Theorem of [7], there is an integer go such that for all integers ~ > go and for all maximal ideals m of R,

~. R n ~ ~ = m~-~o(~. R n ra~" ).

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Put L := t0 + 1. We claim this is the desired bound. Let m be a maximal ideal of R and suppose ~ E m L. Then

h e h . R n m *o+' =m(-~.Rnr. ~o) C~.m.

Since R is an integral domain and h r 0, this contradicts the fact that m is a proper ideal, so h ~ m L. []

Since it may usefully illustrate special properties of the rings K(X), we give an alternate proof of Theorem 4.4 for the case that A is K-affinoid - - call this Theorem 4.4'.

Proof. Clearly, it suffices to prove Theorem 4.4' for the affinoid algebras A = KIX ~. We further reduce to the case that K is algebraically closed by showing that the inclusion K(X) --~ "KIX~ is faithfully flat for any algebraically closed complete valued extension field K of K. Indeed, if it is faithfully flat, then by [17], Theorem 7.3, the induced map S p e c ~ I X ) --* Spec KIX ) is surjective, and by [17], Theorem 9.5, the Going Down Theorem holds. Suppose Theorem 4.4' is known for prime ideals of ~ ( x ) . Let p E Spec K(X), h E K(X) \ p. There are only finitely many minimal elements ~ l , . . . , ~ r of V ( p - F / X ) ) such that P = ~i N K(X). Choose L E IN large enough so that h ~ ~i + J/" for any proper ideal J D ~i , 1 < i < r. Given a proper ideal I of K(X) containing p, we want to show that h ~ p + I t . We may as well take I to be maximal. Then for some J E SpecK'(X), p C I = J N K(X). By the Going Down Theorem and the minimality of ~ t , . . . , ~ r , we have for some i = 1 , . . . , r that ~ i C J. Thus, h ~ ~Ji + jL, which implies h ~ p + I L.

Cla im. K(X) ~ ~ ( X ) is faithfully fiat. (Hence by transitivity, K(X) --~ K'(X) is faithfully fiat for any complete valued extension field K ' of K.)

By [4], Proposition 7.1.1.1, no maximal ideal of K{X) can generate the unit ideal of ~ ( X ) , so by [17], Theorem 7.2, it is only necessary to show that K (X) --* -K(X) is flat.

Let Y be a single new variable. By the ultrametric Weierstrass Preparation Theorem [4], Theorem 5.2.2.1 and Proposition 5.2.4.1, we deduce easily from the flatness criterion of [17], Theorem 7.6, that (i) if K(X)[Y] ---* -K(X,Y) is flat, then g (x, Yl -* -K(X, YI is flat.

Let 9Yt be any maximal ideal of-K(X,Y I and let m := ~Yt fl ~ ( X / [ Y ]. By the weak Nullstellensatz [4], Proposition 7.1.1.3, there is some Y0 e K', ]Y]0 < 1, such that Y - Y0 E m C flit. Taking quotients by (Y - Y0) and applying [17], Exercise 22.3, we deduce the flatness of the inclusion ~(X)[Y]m --* ~ ( X , Y).~ from that of the identity map -KIXImn-~(x) --~ ~' (Xl .~n~(x) . Since this holds

for any maximal ideal flit of ~ ' (X, Y), we conclude by [17], Theorem 7.1, that (/{) K'(X)[Y] --, "K<X,Y) is fiat.

Now we can show that K(Xt , . . . ,Xm) -..* "K(X1,...,Xrn) is fiat for every m E IN. When m = 0, this is clearly true. Suppose K(X) --* "K(X) is fiat. Then

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K(X)[Y] ~ "K(X)[Y] is flat. By (ii) and transitivity, we see that K(X)[Y] .--.* ~ ( X , Y) is flat. Therefore, by (i), the claim follows.

We have reduced the problem to proving Theorem 4.4' for prime ideals p of ~ (X) . Let h e K(X) \ p. The points z e V(p) C ~-m correspond bijectively to maximal ideals m~ D P, where R is the valuation ring of K. Thus it suffices to find an integer L so that for any such maximal ideal m~, h ~ p + m L. Let E C ~ consist of those points x for which h ~ nt>0(P +mtx); by the Krull Intersection Theorem, then, E = V(p) is a zeroset. For each g E IN, let El C R-~

t consist of those points x for which h ~ f30( p . = + mJ~). Then Ut>0El_ = E.

Let fl . . . , fr generate p. The system of equations

T~:h = glT~:fl + ' " + grT~:fr mod rn~ (,)

is linear in the Taylor coefficients of the 9i up to at most order g. The entries of the augmented matrix and the matrix of system coefficients are given by power series which are strictly convergent in x. A point x0 E ~-m belongs to Et exactly when the corresponding equation (,) cannot be solved for the gi. Thus each Et is a finite union of differences of zerosets. After an application of the stationarity argument in Lemma 4.3, we have proved Theorem 4.4'. []

We now apply Corollary 3.4 to obtain a relation between approximately graphic points and divisibility properties of the coefficients of the unique solution G E K((X))[[Z~ to the equation T x f = G o ~x~O.

L e m m a 4.5. Let the map (to, f): R 'n ~ K n x K be given by strictly convergent power series and suppose (~o, f ) satisfies the rank condition. Let 6 be a nonzero (n x n) minor of D~o. Find G E K ( ( X ) ) [[Z~ as in Corollary 3.8, G = ~"]~ P,v~r. Z ~,

p P~q~

so that Txf = Go ~ o .

Suppose Xo E R m is approximately graphic up to order L with respect to the map (~o, f ) . Then

p# E q# �9 K ( X ) + mrL/1ol(zu+x),

where 6 E m u \m u+x and m~ o is the maximal ideal of K (X) generated by X - xo. ~0 ~ XO

Proof. Let m be the ideal of K[[X]] generated by the variables and let 9Jr be the ideal of K[[X, Y]] generated by the variables. By choice of x0,

Txof(X) - g o T~ mod m L

for some g E K[Z]. Thus

T x + . o f ( Y ) ~ ( T ~ ( x + x o ) _ ~ ( x o ) g ) o T ~ + = o ~ O ( Y ) mod ff/t L.

On the other hand,

TX+~of(Y) = Gx+~o o T~

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V" p~(X+~o) Z~" where Gx+~:o : - ~,~!~(x+~o) Note by Corollary 3.4 that

6(X + x0) 21~1 p/3(X + zo) K(X). + xo)

PutH(Z) : - (Tv (x+ ,o ) - v ( , o )g )=~ ,~ ,X . z ' . W e h a v e

S o T~ -- Gx+xo o ~x+~o~ mod ~ L . (,)

l <_i,j <n Function Theorem as in Lemma 3.2, we obtain a unique ~' E K((X)) [y~m with ~(0) = 0 such that

o = ( Y 1 , . . . , Y , )

and such that

~(~x+~:o~l,.'. ,T~+xo~n,Yn+l,...,Fro)= Y. (**)

Corollary 3.4 and (**) together imply that

~P(6(X + Xo)~Y1,..., 6(X + Xo)2Ym) e (K(X))[[Y]].

Composing on the right in ( ,) yields

H o T~ o ~P(5(X + xo)2Y) _= ax+,o o T~ o ~(5(X + xo)2Y) mod ~'~L.

Hence

H(6(X + xo)2Y1,... ,6(X + x0)2Y,) -

- Gx+~:o(6(X -}- x0)2y1, . . . , 6(X + xo)~Yn) mod ~ L .

Comparing coefficients of like powers of Y yields

5(X + x0)2[~lhp -- ~f(X + xo) ~]plp~(X + xo) mod m L-I~].

Cancelling common factors and reducing the order of the congruence appropriately, we obtain

p~(X + xo) "= ql3(X + xo)hz(X) mod m L-IpI(~'+I).

The lemma follows. [:] We now apply Theorem 4.4 to obtain an estimate on the bound L of

Lemma 4.3. Let to, f , 6, G, p~ and qz be as in the statement of Lemma 4.5. By Theorem 4.4

(or otherwise) there is some ~ E IN such that 6 ~ I 7r for any proper ideal I of K(X) . Let 61, . . . ,6r be the prime factors of ~f. Define integers LiB, 1 < i < r, /~ 6 IN", as follows. If 6~ does not divide q~, put LiZ := 0. If 5~ does divide q#,

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then by Corollary 3.8, 8i does not divide PE" Hence by Theorem 4.4, there is an integer LiE such that

p~ ~ 61 �9 K ( X ) -t- rn L'~-IEl(27z+x)

for every z E V(6i) C R rn.

L e m m a 4.6. (In the preceding notation.) Put L := maxi min E Lip. Then any point xo E R m which is approximately graphic up to order L with respect to the map (~, f ) is also graphic.

Proof. Suppose x0 E R m is non-graphic. Then by Corollary 3.8, for some i, 1 < i < r, 61(xo) = 0 and for some fl E INn, 61 divides q#. Find such a f l with

L LiE minimal, so [ill < z~-TT. Suppose x0 is approximately graphic up to order

L. Then by Lemma 4.5, PEE qE "K(X) +mLo IEl(z~+l)" Since L > Li# and since ~i divides qE, this is not possible. []

The following Approximation Theorem is now a straightforward consequence of either of Lemmas 4.3 or 4.6. We use this Approximation Theorem in the proof of Lemma 5.5.

T h e o r e m 4.7. Let ~1 , . . . , ~ n , f E K ( X ) , where ( ~ , f ) satisfies the rank condition. Then there is an integer L E IN such thai for every xo E R m and every g >_ L, i f f o r s o m e f f e K[Z1 , . . . ,Zn]

T~of(X ) "~ y o T~xo~(X ) mod (X) l, (4.7.1)

then there is a unique g E K{Z} such that

Tzo f (X) = g o T~ (4.7.2)

furthermore, g = y mod (Z) b (4.7.3)

for any b with b(Iz + 1) < ~, where ~ is any nonzero (n x n) minor of D~ and 6(x + e (x ) . \ (x ) . +1

Proof. Find L E IN as in Lemma 4.3 or 4.6. If l > L and (4.7.1) holds for some y E K[Z], then by Lemma 4.3 or 4.6, there is some g E K { Z ) such that (4.7.2) holds. The uniqueness of this solution follows from Lemma 3.2.

It remains to verify (4.7.3). Without loss of generality, we assume that ~(0) = 0 and x0 = 0. Let df be a nonzero (n x n) minor of D~. Find ju E IN such that

e (X) ~ \ (X) ~+1. The following claim completes the proof.

Cla im. Let l E IN and g E K[TZ]]. If

go~_-__0 m o d ( X ) l

then g ~_ 0 mod (Z) b whenever b(/~ -b 1) < s When t _< 1 there is nothing to show. Induct on s By the Chain Rule,

64 Robinson

OXi(g~176176176 OXi m ~ l < i < m .

By Cramer's Rule, this implies

Og OZj o~o=0 m o d ( X ) t-("+l) , l < j < n .

By induction, for every j = 1 , . . . ,n, ~ = 0 rood (Z) b-1 when b(p + 1) < ~. Since the characteristic of K is 0, the claim follows. []

5. S m o o t h po in t s a n d d i i f e ren t i ab i l i t y

The results in this section on smooth points and differentiability will follow from the results of Sections 3 and 4 together with the desingularization result of Theorem 5.2. Theorem 5.2 follows from results of [6] and from Hironaka's embedded resolution of singularities.

Def in i t i on 5.1. (Denef-van den Dries [6].) Fix a prime number p 6 7/. For n 6 IN, let Pn be the set of nonzero n th powers in the p-adic field Qv and let -P,, := P , U {o}.

m m (i) A subset S C 7/p is called s e m i a n a l y t i c a t a po in t z0 6 71p iff there is a polydisk U containing x0 such that U f'l S is a finite union of sets of the form

{y 6 V ] f(y) = 0,gl(y ) ~. P,~,... ,gr(Y) 6 P,,},

where f , g l , . . . , gr are functions given by power series convergent in U with coefficients in Qp.

(ii) A subset S C 71~ is called s e m i a n a l y t i c iff S is semianalytic at each point of 71~.

m IT~ (iii) A subset S C F/p is called s u b a n a l y t i c a t a po in t z0 6 7/p iff there is an open neighborhood U of z0 in 71p , an integer M 6 IN and a set

S' C U x 71M, semianalytic in F/~ +M, such that U fl S -- 7r(S'), where

7r: U x 71M ~ U is the projection on the first factor. (iv) A subset S C 7]~ is called s u b a n a l y t i c iff S is subanalytic at each point

of 71~. (v) Let S C 71~. A function f :S ~ 71M is called s u b a n a l y t i c iff its graph is

a subanalytic subset of 7/~ n • 7/M.

The following desingularization theorem is a p-adic analog of Hironaka's Uniformization Theorem for real subanalytic sets. Due to the lack of a suitable Fiber-Cutting Lemma (see, e.g., [3], Lemma 3.6 or [12], Lemma 3.7.2,) we are as yet unable to establish the interesting equality dim M = d i m s for some compact Qfana ly t i c manifold M, as below.

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T h e o r e m 5.2. Let S C 71'~ be closed and subanalytic. Then there exists a compact Qp-analytic manifold M and an analytic map h: M --~ 77~ such that S = h(M).

(For the definition of Qp-analytic manifold and Qp-analytic map, see [20], LG Chapter 3.)

Proof. After an application of Denef and van den Dries's Uniformization The- orem [6], Theorem 2.4, we may assume S closed and semi-analytic. By [6], Corollary 3.40, we may even assume that S is a set of the form

{= E ~7~ n I gl(=) e ~ n , , . . . , g r ( z ) E ~nr} ,

where g l , . . . , g~ E Qp(X). After multiplying each gi by a suitably small scalar from Qp, we may assume each [[g/[[ <-- 1. Therefore, S is the image under projection on the first m coordinates of the zeroset

~ . + r I g l (x) = u; ' ,gr(x) = u."'}. ( (=, u l , . . . , u.) e - p , . . .

It is therefore no loss in generality to assume S is a zeroset. To complete the proof, it now remains only to apply the following version of

Hironaka's theorem on embedded resolution of singularities cited in [6], Theorem 2.2.

T h e o r e m . (Hironaka.) Let f l , . . . , f~ G Q p ( X l , . . . , X,~) \ {0}. Then there exist a compact Qp-analytic manifold M of dimension m and a surjective analytic map h: M --* 7], such that:

( i ) For every b G M there exist local coordinates (Y l , . . . , Ym) at b such that ul(b) = . . . = u~.(b) = o and

f i o h = u i y ~ " . . . y ~ m ' ' , l < i < r ,

in an open neighborhood U of b, where the ul are analytic in U, ul never vanishes on U and Nij E IN.

(i i) h is a composition of finitely many blowing-up maps with respect to closed submanifolds of dimension < m - 2 (contained in the preimage ofU, fi-~(O).)

[]

In [12], Definition 3.1, Hironaka defined the class of subanalytic subsets of a given real-analytic space to consist of those subsets which, locally about every point of the space, can be written as a finite union of differences of images of proper real-analytic maps. As a first application of Theorem 5.2 and the analytic elimination theory of Denef and van den Dries, we show that the analogous statement yields a definition of the class of subanalytic subsets of 7]~ equivalent to the one given in [6].

C o r o n a r y 5.3. A set S C 77'~ is subanalytic if, and only if, there are:

66 Robinson

O) compact Qp-analytic manifolds M1, . . . , Mr, Oi) closed analytic sets ~ C Mi, 1 < i < r, and

Oii) Qp.analytic maps f i ,g i :Mi "* 7]~, 1 < i < r

such that:

i =1

Moreover, we may even take �88 = M i in (*) .

(,)

Proof. Using the compactness of 7/v and of the manifolds M1,. �9 Mr, it follows directly from the analytic elimination theory of Denef-van den Dries (in particular, from [6], Corollary 1.6) that any set S C 7/v of the form (*) is subanalytic.

Suppose S C 7iv is subanalytic and let d := dimS. We show by induction on d that S may be represented in the form (*). When d = 0, S is finite by [6], Corollary 3.26, and we are done. Let d > 0 and suppose each subanalytic subset of 7iv of dimension less than d has a representation of the form (*), where each t~, in addition, is the compact Qp-analytic manifold Mi.

Let LDn be the formal language described in [6]. By [6], Corollary 1.6, any set definable over 7iv by an LDn-formula is subanalytic, and conversely. Hence, since

S is subanalytic, so are :~ (the closure of S in 7/v), ~ \ S, ( ~ \ S), and any set obtained from these by Boolean operations. By [6], Corollary 3.26, Theorem 3.14

and Lemma 3.16, d i m s = d im~ = d, d i m ~ \ S = d i m ( ~ \ S) < d and d i m S n

\ s) < d. Now apply Theorem 5.2 to write

-~ = I ( M ) , (-~\ S) = g(N),

where M and N are compact Qv-analytic manifolds and f and g are Qv-analytic maps. It is no loss in generality to assume M = N: take the Cartesian product

and extend f and g appropriately. Since dim S gl (S \ S) < d, by induction, we h a v e

s n s) = U \ i=1

for some compact Qp-analytic manifolds M1, . . . ,Mr and ~v-analytic maps fi , gi: Mi ---* 7] v , 1 < i < r. Since

we have

S = f ( M ) \ g ( M ) U 0 f i (Mi) \ g i (Mi ) , i=1

which is a representation of the desired form. r~

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Given a closed subanalytic subset S C 7]~, Theorem 5.2 gives us a compact manifold M and an analytic map h: M ---* 77~, such that S = h(M) . Since M is compact, we may think of h as s maps from 77~, each given by strictly convergent power series. To find the smooth points of S, we need to have a fiberwise version of Definition 3.1 and we need to be able to handle s maps at a time.

D e f i n i t i o n 5.4. Suppose that (~pl, f l ) , . . . , (~ , , f , ) : R m _._, K" x K r are given by strictly convergent power series and tha t ( ~ , i f ) satisfies the rank condition for each v = 1 , . . . , s. A point Y0 E K " x K r is said to be a p p r o x i m a t e l y s- fo ld f i b e r w i s e g r a p h i c u p to o r d e r t with respect to (~v 1, f l ) , . . . , (~ , , f , ) iff there is a fixed g = ( g l , . . . , gr) with g l , . . . , gr E K[Z ~ , . . . , Z,,] such that for each tJ = 1, . . . , s and each zo e (~ov, f v ) - l ( y o ) ,

TroIV(X) =_ g o T~zog~v(X) mod m t.

If, for some g = (g~, . . . ,gr) with g~, . . . ,gr E K { Z } , actual equality holds in the above, then we call the point Y0 an s - fo ld f i b e r w i s e g r a p h i c p o i n t of (~ol fl),...,(~ps,fs).

We next prove a fiberwise version of Theorem 4.7.

L e m m a 5.5. Suppose that ( r ~ K n • K r are as in Definition 5.4. There is some L E IN such that for any y E K n x K r, if y is approximately s-fold fiber'wise graphic up to order L then y is s-fold fiberwise graphic.

Proof. For each v = 1 , . . . , s, write f v = ( f { ' , . . . , f~'). By Corollary 3.8, there are elements p[#, q~f3 e K ( X ) such that

T x I : = o

where Pb za

#

Furthermore, we may assume for each fixed u = 1 , . . . ,s and i = 1 , . . . ,r that conditions (3.8.1) and (3.8.2) are satisfied with respect to the map ( ~ v f : ) .

For each /~ ,v = 1 , . . . ,s, let luv be the ideal of K ( X , Y ) generated by all elements of the form

p S ( X ) q S f Y ) - p S ( Y ) q S ( X ) ,

where 1 < i < r and ~ E IN n. Since K ( X , Y } is Noetherian, there is some r E IN such tha t for every l~,u = 1 , . . . ,s, Iuv is generated by the elements p • ( X ) q S ( Y ) - p • ( Y ) q S ( X ) , 1 < i < r, 181 < t l .

By Theorem 4.7, there is some ~2 fi IN such that for every v = 1 , . . . , s and i = 1 , . . . , r and every z E R m, if z is approximately graphic up to order ~2 with respect to (~", f~), then z graphic.

68 Robinson

For each v = 1 , . . . , s , by the condition on D~ ~, we may choose an (n x n) minor 5 v of D~o V which is not identically zero. By Theorem 4.4, find p E IN such that 61, . . . ,6 ' f~ I t for any proper ideal I of K ( X I.

Let L > t 2 , t l ( p + 1) and suppose Yo 6. K n x K r is approximately s-fold fiberwise graphic up to order L. Then for some g = (g l , . - . ,gr) with gl . . . . ,gr E K[Z], for each v = 1 , . . . , s and for every z0 e (~o ~, f~) - l (y0) ,

T ~ o f ' ( X ) - g oT~ mod m L. ( . )

Since L > s this implies for v and z0 as above that G ~. E K { Z } and that

T " . o l ( x ) = a L o

G / / // // ' where G~xo := ( 1~o, . . . , G~o) and Gix ~ := Z ~.

Let s0 e (~", f " ) - l ( y o ) and $1 e (~v, f~ ) - l ( yo )" Then by the above equation and ( . ) , Theorem 4.7, equation (4.7.3) implies that all the generators of I~u vanish at the point (z0,Zl) . Hence G~o = G~ u, for each/~,v = 1 , . . . ,s and any x0 e (T~ , f~ ) - l (y0 ) , Zl e (~ou,fv)-1(yo). From this, we conclude that Y0 is s-fold fiberwise graphic, as desired. []

Let S C 27~+~ be a subanalytic set of dimension n. Then S is said to be smooth at Yo E S iff there is a neighborhood U of y0 such that S fl U is an n-dimensional submanifold of 7/n+~ We find it convenient, however, to give the following equivalent definition. In order to relate this to the approximation results of Section 4, we also give a definition of smoothness up to order L The reader is reminded, by way of motivation for the following definition, that , over Qp, the relation between differentiation and Taylor approximation is problematic. (See the remarks following Defnit ion 5.6.)

n De f in i t i on 5.6. Let U C Qp be open, let z0 E U and let f : U --+ Qp. Then f is said to be s t r o n g l y d i f f e r e n t i a b l e u p to o r d e r s a t x0 iff there are: some neighborhood U' of z0 in U, a function e: U ~ -+ IR, continuous at x0, with e(x0) = 0, and constants a s E Qp for every fl E IN" with [ill < s such that

If(z) -- ~ a/~(x-- < r Ix- XO) '0 X01 s

I#l<t

for every z 6 U'. Let f = ( f l , . . . , f r ) : U --.+ Qrp. Then f is said to be s t r o n g l y d i f f e r e n t i a b l e u p to o r d e r t a t z0 iff f l , . . . , fr are each strongly differentiable up to order t at z0.

For any n-tuple i = ( i l , . . . , i n ) with 1 < il < .-" < in < n + r, denote by rt the coordinate projection . - ~ + r n. t."p When r = 0 note that ~r| is just the identity map.

7/n+r be a subanalytic set of dimension n, let y0 E S and let s 6 IN. Let S C - p The set S is said to be s m o o t h u p t o o r d e r t a t y0 iff there is some n-tuple i as above and some neighborhood U of Y0 such that

(i) 7ri(S n U) C ~7~ is open,

Robinson 69

(ii) r i [snv: S n U -* ~rl(S V~ U) is a homeomorphism, and rt lsnu is strongly differentiable up to order t at lri(y0). (iii) -1

We say that S is s m o o t h at Y0 iff, in addition, we have

( i v ) -1 7rilsn U is analytic at ri(y0).

When r = 0, it is equivalent to say that Y0 is in the interior of S, that S is smooth up to order ~ at Y0, for some g _> 0, or that S is smooth at Y0.

The s i ngu l a r locus Sing S of S is defined to be the set of points y E S such that S is not smooth at y.

In the real variables case, by Taylor's Theorem, if a function is differentiable up to order ~+2 at z0 then it is strongly differentiable (in the obvious sense) up to order ~ at z0. The corresponding assertion is false, in general, for functions f : U --~ Qp. (See Schikhof [18], Example 26.4, for a function : 27p ---, 27p which is everywhere differentiable up to every order but not strongly differentiable up to order 2 anywhere.) It is clear that ~p-analytic functions are strongly differentiable up to any order, and it seems reasonable to conjecture, moreover, for functions subanalytic over ~p, that there is a relation as in the real variables case between ordinary and strong differentiability.

Note, by the stratification result of [6], Theorem 3.14, that the set of smooth points of a subanalytic set is non-empty.

By the above definitions, after desingularizing, Lemma 5.5 becomes the following.

L e m m a 5.7. Let S C 7]~ +r be a subanalytic set of dimension n. There is some L E IN such that for every y E S, if S is smooth up to order L at y then S is smooth at y.

Proof. When r = 0, just take L = 0. We assume from now on that r > 0. For each n-tuple i as in Definition 5.6, let SI denote the set of points y E S

such that the coordinate projection 7r|: 71~+r _., 7]~ induces a homeomorphism of some neighborhood of y in S onto an open subset of 7/~. The lemma reduces to showing that there is some L E IN such that for every y E Si, if there is a

71" - 1 neighborhood U of y such that ils~nv is strongly differentiable up to order L at 7q(y), then it is analytic at 7ri(y).

Let LDn be the formal language described in [6]. By [6], Corollary 3.6, any set definable over 7]p by an Lgn-formula is subanalytic, and conversely. It is clear

7/n+r --~ n that the condition 'y E S and the coordinate projection ~ri:--p ~p induces a homeomorphism of some neighborhood of y in S onto an open subset of 7]~' may be translated into an Lgn-formula. Hence, SI is subanalytic. Similarly, the closure ~l of Sl is subanalytic, and, if Si • g, by [6], Corollary 3.26, dirndl = n. Note that Si is relatively open in ~l.

Since ~i is closed and subanalytic, we may apply Theorem 5.2. Thus there are maps h 1 . h s : 2Z~ n --* 7/n+r , . . , _p , each given by strictly convergent power series, such that

70 Robinson

= h' (z ' ; ') u . . . u

Since d im~i = n, by the Rank Theorem, for u = 1 , . . . , s , rank Dh v < n. By j denote the r-tuple j = ( j x , . . - , j r ) with 1 < Jt < "'" < jr _< n + r and {ix,. , i , ; j x , . . ,Jr} = { 1 , . . . , n + r ) . B y r j : 2 v "+r r . . . . p ---, 7]p denote the projection on the corresponding coordinates. For u = 1 , . . . , s, put

~o u : = ~ l o h u and f v : = T r j o h u.

If the generic rank of ~ou is less than n, then dim a'l o hU(T]~) < n, so that Is ~Tt ~i = Uv#uh (Tip). We therefore assume for u = 1 , . . . , s that the rank condition

holds for ( ~ , f~). Let L be as in Lemma 5.5. Since hX, . . . , h ~ are closed maps, it follows from

the definition of strongly differentiable that L is the sought after integer. [3 Since a function is strongly differentiable up to order L, respectively analytic,

about a point in the interior of its domain if, and only if, its graph is smooth up to order L, respectively smooth, over that point, we immediately deduce the following.

n T h e o r e m 5.8. Let S C Tip and suppose f: S ---* 71rp is subanalytic. There is an L E IN such that, for every point x in the interior of S, f is analytic about x if, and only if, f is strongly differentiable up to order L at x.

7/n+r be a subanalytic set of dimension n and let y E S. The Let S C --p s t a t e m e n t ' S is smooth up to order L at y' depends only on finding the finitely many coefficients of a Taylor approximation up to order L for the inverse of some coordinate projection from a small neighborhood of y in S onto some

n open set in 77p, whence this s tatement may be translated into an LaDn-formula. Therefore, by Lemma 5.7 and [6], Corollary 1.6, we obtain the following.

7/n+r be a subanalytic set of dimension n. The set of T h e o r e m 5.9. Let S C - p points y E S such that S is smooth at y is itself a subanalytic set of dimension n. Hence Sing S is subanalylie as well.

R e f e r e n c e s

1. Bartenwerfer, W.: Die Beschr~nktheit der Stfickzahl der Fasern K-analytischer Abbildungen. J. reine angew. Math. 416, 49-70 (1991)

2. Berger, R., Kiehl, R.,Kunz, E. and Nastold, H-J.: Differentialrechnung in der an- alytischen Geometrie (Lecture Notes in Math., 38) Springer (1967)

3. Bierstone, E. and Milman, P.: Semianalytic and subanalytic sets. Publ. Math. IHES 67, 5-42 (1988)

4. Bosch, S., Gfintzer, U. and Remmert, R.: Non-Archimedean Analysis, Springer (1984)

5. Bourbaki, N.: Alg~bre, Masson (1981) 6. Denef, J. and van den Dries, L.: p-adic and real subanalytic sets. Ann. Math. 128,

79-138 (1988)

Robinson 71

7. Duncan, A. and O'Carroll, L.: A full uniform Artin-Rees Theorem. J. Reine Angew. Math. 394, 203-207 (1989)

8. Fresnel, J. and van der Put, M.: G~om6trie Analytique Rigide et Applications, BirkhRuser (1981)

9. Gabrielov, A.: Projections of semianalytic sets. Fuuct. Anal. and its Appl. 2, 282- 299 (1968)

10. Gabrielov, A.: Formal relations between analytic functions. Math. USSR Izvestija 7, 1056-1088 (1973)

11. Hardt, R.: Stratification of real analytic mappings and images. Invent. Math. 28, 193-208 (1975)

12. Hironaka, H.: Subanalytic sets, in Number Theory, Algebraic Geometry and Com- mutative Algebra, in honor of Y. Akizuki, Kinokuniya 453-493 (1973)

13. Lipshitz, L.: Isolated points on fibers of affinoid varieties. J. reine angew. Math. 384, 208-220 (1988)

14. Lipshitz, L.: Rigid subanalytic sets. Amer. J. Math. 115, 77-108 (1993) 15. Lojasiewicz, S.: Ensembles semi-analytiques, IHES (1965) 16. Malgrange, B.: Frobenius avec singularit6s, 2. Le cas g~n~ral. Invent. Math. 39,

67-89 (1977) 17. Matsumura, H.: Commutative ring theory, Cambridge University Press (1989) 18. Schikhof, W.: Ultrametric calculus, Cambridge University Press (1984) 19. Schoutens, H.: Approximation and subanalytic sets over a complete valuation ring,

Ph .D. Thesis. Katholieke Universiteit Leuven (1991) 20. Serre, J-P.: Lie Algebras and Lie Groups, 1964 lectures given at Harvard Univer-

sity, Benjamin 21. Tamm, M.: Subanalytic sets in the calculus of variation. Acta Math. 146, 167-199

(1981)

Zachary Robinson Department of Mathematics Purdue University West Lafayette, IN 47907 USA

This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990.

(Received Noventber 24, 1992)

smooth points of p -adic subanalytic sets

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