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An extension of whitney’s spectral theorem

An extension of whitney’s spectral theorem by J.-G1. TOUGERON i. Notations and results. For jye'R13 and YcR^ \y\ denotes the euclidean normof> and rf(j/,Y) the euclidean distance from y to Y. If Y is empty, we write d{y, Y) == i. Let ^ denote an open set in R^ and ^{O.p) the R-algebra of all G00 real-valued functions in Qy. When j^eQp, let ^m denote the R-algebra of Taylor expansions of order m atj/ of all elements in <^(t2p); if w<oo, ^w is isomorphic to the algebra ^p/m^4'1, where m? denotes the maximal ideal of the formal power series ring ^'=^[[^1, . . . , y ]] $ if m=+co, ^m (simply written^) (1) is isomorphic to ^ (by the generalized Borel theorem). Let T^ : S{^Y-> [y^Y denote the projection associating to each function G its Taylor expansion of order m at y. If Y is a compact set in 0. we write |G[^= = sup [D^GO^I. We provide ^(ti^with its usual structure of a Fr^chet space, !/£Y |fc|^w defined by the family of all semi-norms Gh>|G|^, where Y ranges over the set of compacts in 0,y and me'N. Let M be a submodule of ^(ftyY and let us write M={GG^(^|Vj^,3G'eM so that G-G' isflatatj/}= 11 (T^-^T.M). According to a standard result of Whitney (B. Malgrange [i]), M is the closure M of M in ^(tip)^: we propose to extend this theorem. Let <I> denote a G00 function from an open set Q^ in Rn to 0. . The mapping O defines a homomorphism of R-algebras <S>*: S(ft^)3g h>^o(pe<;?(^J. Let Y be a 0*-homomorphism from ^(^)3 to <f(0.n)\ i.e. Y is a homomorphism of abelian groups and, VGe^(Q^ and V^e<?(^): V{g.G)=(S>\g) .Y(G) . For ye^ and ^eO-1^), the mapping Y induces an R-linear mapping Y^ : [^Y -> (e^)', so that T^oY==Y^oT^. For XcO"^^), we note Y^ the R-linear mapping (^y^^Vn-(T^(V))^^x6 n (^rw)r. Finally, let T^ be the mapping ^(^n)r^F^(T^F)^x^ H (^T1)'- fl;ex 3? £ X . We propose to determine the closure Y(M) of Y(M) in ^{f^nY- Therefore, let us write ^^^{Fe^tiJ^Vj/e^, 3GeM such that Y(G)-F is flat on Y-^j)} -.ro,^-1^'1^-1^0^^- (1) We shall omit afterwards the index m, if w == + oo, and shall write: Ty,^, ... instead of T°°, T?, ... 18* 140 J.-CL. TOUGERO N We shall prove the following result: Theorem (1.1). — Let us suppose that 0 verifies the following condition: (H) For all compact sets Xc^ and Yc^Qp, there exists a constant ocj>o such that, Vj/eY: roo= sup (^<D-l(^))a/[(DW-^|)<oo. a^exvo-^y) T^ Y(M)==Y(M). It is easy to find G00 mappings 0 which do not satisfy this condition. Nevertheless, we shall prove the following result: Theorem (1.2). — An analytic mapping 0 verifies the condition (H). Both following paragraphs are devoted to the proofs of these theorems which are independent of each other. In the last paragraph, we give a refinement of the Theorem (1.2), when 0 is a polynomial mapping. 2. Proof of theorem i . 2. Definition (2.1). — Let 3 be a finitely generated ideal of a subring of the ring of germs at x° in K^ of continuous functions with real values. Let 91(^)3 . . ., <pg(^) denote real valued functions, continuous in a neighborhood of x° and such that their germs at x° generate 3. Let V(3) be the set of their zeros. We say that 3 verifies a Lojasiewic^, inequality of order a^>o (or simply that 3 verifies J2^(a)) if there exist a constant G>o and a neighborhood V of x° such that, V^eV, Sly.M^G.^V^))". i==l Let iQp be an open set in R7', ^ an open set in If, j/=(j^, ...,jp) and x=(x^y .. ., x^) coordinate systems in Q.y and ^ respectively. Let (9 be the sheaf of germs of analytic functions with real values on Q.^x0.p; ^ a sheaf of ideals, analytic and coherent on Q^xOp. For (^°, y°)e 0.^x0. „ , we denote ^yo) the stalk of ^ at the point {x°,jy°). Let 9^, . . ., cp, be generators of the ideal ^^y we denote ^,1/0) the ideal generated by (pi^,./), ...,(p,(^,j/0) in the ring O^y^ of germs at {x0,^) in Q^x{^°} of analytic functions with real values. Permuting x and y, we define similarly the ideal -^yo) of ^\y°}' Finally, let V(J^) be the set of zeros ofJ^. Theorem (1.2 ) is an easy consequence of the following one (Lojasiewicz inequality with a parameter): Theorem (2.2). — Let X be a compact set in 0.^, Y a compact set in £ip. There exists a^o such that the ideal J^ verifies J^(a), V(^,j/)eXxY. Indeed, let us suppose this theorem is true, and let 0 be an analytic mapping. Let ^ denote the analytic and coherent sheaf generated on O.^x0.p by ^iM—j^, .... O {x)—jy . Let X, Y be compact sets in ^, Q.y respectively. By (2.2) applied to J^, V(,y°,^) 6=XxY, there exists a constant G^y)>o such that for ^ in a neighborhood of 24^ AN EXTENSION OF WHITNEY'S SPECTRAL THEOREM 141 ^rIOM—j/I^G^^.^O- 1^)) 01 . Hence, the set X being compact, there exists a constant Gy>o such that, V^eX: lOW-j^C^O-1^))-. Clearly, condition (H) follows. Proof of (2.2). — Obviously, condition JSf(oc) is verified, with a==o, for (^,j/)^V(J^). The set XxY being compact, it suffices to find, for (^.j^eV^), an oc;>o such that J^y) verifies JSf(a) for (A;,J^) in a neighborhood of (A;0,^0). We shall suppose that {x°,y°) is the origin of B^xR^. Now, it is enough to prove the following result: (2.3) There exists an a^o such that ^^ verifies JSf(oc) for (o^eV^) and \y\ small enough. Indeed, let 9i(^), . . ., <pg(A:,^) generate ^ in a neighborhood of (o, o), and let us consider the sheaf ^ generated on a neighborhood of the origin of R^xB^xR^ by 9i(^+^), . . ., 9,(x+^,^). By (2.3) applied to the sheaf / (with the para- meter {^y} instead ofj/), there exists an ocj>o such that ^^=J^^ verifies .JSf(oc) for [^y ) in a neighborhood of the origin. Proof of (2.3). — We proceed by induction on the height k of the ideal A) Q) . There exist sheafs of ideals ^>1, . . ., ^r, analytic coherent on a neighborhood of the origin of R^R^, such that <^o)? • • "> %o) are prime ideals of height J>/;, and an integer P^i, such that: ^D(^ln...n^r)p. Clearly, if ^j^ verifies J§f(a,) for y small enough, e^o ^ verifies J§f(p S a,) forj/ small enough. Hence, we may suppose that ^[0,0} ls prime and its height equals k. Let <p(j/) be analytic in a neighborhood of the origin of oxR^ and null in V(J^) n (o X R^) in a neighborhood of the origin. Let / be the analytic coherent sheaf on a neighborhood of the origin of R^R^, generated by ^ and 9: obviously, ^,y}==^,y] forj^ small enough. If cp(^J^, we get ht /^Qy>k and hence the result is proved by the induction hypothesis. Therefore, we may suppose that cp(=eXo 0)5 i-e. ^o)3-^) a^ -%o) ls the ^eal of germs <p(j;) null in V(A) n (oxR^) . Lemma (2.4). — 14Wz ^ preceding hypothesis, let k—l be the height of the prime ideal -%o)- After an eventual permutation on the coordinates x^, ...,^, there exist <pi, ... , 9^^0,0) such that ^= yl? " > ? ?^ ^0,0)' -u ^1 5 • • • 5 ^J Proo/'. — We proceed by induction on the height k of e^o ^. Let us suppose that k>{. There is a sequence (o,Y)eV(J^), V-^o, such that for each i: ^ .4=0 (otherwise ^^ would be generated by ^,o))- After an eventual linear change of coordinates on the variables ^, ...,A^ , we know (following the analytic preparation theorem, Malgrange [i]) that there exists, for each i, a distinguished polynomial T^=^i+^,^(^J;).^^~l+...+^.,^(^,J;)£^o,^ (we write x'=(x^...,x ^ and the a^ 141 I^ J.-CL. TOUGERO N are analytic functions of {x',y) in a neighborhood of (o,y)). Besides, we may suppose that ~»~^^W (Ir^cd, there exists a smaller integer B,>o such that ^_^4-J •• 1 a3,w ftc3'4'1 ' t0-!'1)' we have only to substitute ——— for Y,.) Hence, there exists y^ „, such that 8<pl d- <r 1 ^ (oto)- Let Q' be the sheaf of germs of analytic functions with real values on R^xR^^jiQ eR-xR^l^o } and let us write ^'=^n 0'' . There exists an integer ^ such that htJ^.)=A: for i^g; besides, <P((», ,,) | ^o,».) is a finitely generated module over ^(o^Ko.i,.) and hence their Krull dimensions are equal (by the Gohen-Seidenberg theorem, Malgrange [i]; th. (5.3), chap. Ill); therefore ht^^=k-i for ^. Since ^ is prime, ht ^ o) = ht J^,, for (• large enough, so that: ht^=k-r, finally, J^op.-%o)- Applying the induction hypothesis to the sheaf Jf '(after an eventual permutation on the variables ^, . .. , A-J, we see that there exist <pa, .. ., y^e^ o, such that ^""J^^o)- Hence: 1^^2, . . ., A"/J P^ ...,y^_8cp i D(<pa, ...,y^) ^ ^ D(^,...,^ ) ^•D(^,...,^)^(0.0'- Since ht^<,)=/:-/ and J^, is prime, there exist y/^, ...,y,e^^, such that, after an eventual permutation on the coordinates Vi, . .. y : ,_P(<P^l, ••-,9, ) . D(yi,...,y,) s2 D(^,...^_/)^o,o>, hence -D(^ .. ., ./,^ .. ..^) = ^•^0,0). By the jacobian criterion for regular points, the localized ring (^g g,) is regular of dimension k and its maximal ideal is generated by y,, .. ., ^. Hence"t'here exists S3^(o,oMo,o) such that: ^3-^0,0)^^1, ••-,%) . Let S be analytic in a neighborhood of (o, o)€R"+p and inducing the germ ^.^3 at the origin. Let / be the sheaf of ideals generated, on a neighborhood of the origin of R^, by ^ and ^. For (o,y) eV(^), y small enough: — There exists a>o such that ^o%)=^o",)+^.%,) verifies ^(oc) (because nt ^0,0) ^t^o.o) ''•"d we apply the induction hypothesis). — ^•^,y) is contained in the sub-ideal ofJ%y, generated by y^, . .. , ^. — Finally, ^ belongs to the ideal generated in % .1 by the jacobian D^1' ••• > ^ ) D(^, ...,^ ) • So Theorem (2.3) is an immediate consequence of the following lemma (Tougeron and Merrien [2], prop. 3, chap. II): Lemma (2.5). — Let 3 be a finitely generated ideal of the ring ^ of germs at the origin in R" of 0°° functions with real values. Let y,, . . ., y/e3 and ^ belonging to the ideal generated 142 AN EXTENSION OF WHITNEY'S SPECTRAL THEOREM 143 in < by (pi, . . ., ^ and all the jacobians ^1? '"? ^ , jo that ^. 3 C (<pi, . . ., (p^). 77^ -U(A^, . . ., ^) ^ 3^3+S.^ verifies JSf(oc), ^ zY^ 3 verifies oSf(sup(2oc, a+i)). J?^rA; (2.6). — Let 0 =(d>i , . .. , Op) be a G00 mapping from ^ to Qy. Let S be the sheaf of G00 functions with real values on ^ (or Q^,); let ^ be the sheaf of ideals D($i $ ) generated on ^ by all the jacobians _ , ' ' ' ?—^-: the set VfJQ of zeros of ^ is the D(^i, ...,^) set of singular points of the mapping 0. Let us consider the following condition: (H') V^eV(^), <^/^ is {by 0) a module of finite type over the ring €y [we set y==<S)(x)), i.e. by the Malgrange preparation theorem (Malgrange [i]): (C/^^W^-^/^+m,.^) is a real vector space of finite dimension (nty: maximal ideal of S). The condition (H') is a very strong one; nevertheless, it is a generic one, i.e. it is verified on an open dense subset of the space of G00 mappings from Q^ to Q. , this space being provided with the Whitney topology. Besides, (H') implies (H). Indeed, let X and Y be compact sets in ^ and ^ respectively. By hypo- thesis, there exists an a>o such that, V^j^eXxY, the ideal generated by Df O $ ) °iW-J^ • • •. ^pW-Vp and all the jacobians _ 1? " > ? p/ in S^^ (ring of germs U(^3 . . ., X^y) at {x°,y°) in R^V} ofG°° functions with real values), verifies J§f(a). By Lemma (2.5), the ideal generated by OiW—j^, . . ., O^)—^, in <^yo) verifies ^(a'), with an a' independent of the point (A^j/^eXxY. Clearly, the condition (H) follows. 3. Proof of theorem 1.1. With the notations of § i, we must show that: ^(M^^M). (3.1) We have: T(M)cY(M). Let FeT(M) and let ye0.y. A finite subset X^ of O"^^) will be called m-essential (m is a positive integer), if ker Y^-i^ = ker T^; clearly, there always exist w-essential sets X^ such that card X^< card {^Y. Let X be a finite subset of (S)~l{y) containing such an X^. By hypothesis, T^F is in the closure of the finite dimensional real space Y^(T^M), and therefore belongs to it. So there exist G^ with G^eT^M such that: T^F==y^(G^) and T^F=Y^(GW). Obviously, G^-G^eker Y^=ker Y^$ thus: T^F^Y^G^), and X being arbitrary : T^-i^F^T^-i^G^. So, WW=(y^-x^)-l(T^-x(y)F)nT^M is a finite dimensional and non empty affine space. The inverse limit W^imW^ is then non empty and contained in UmT^M=T^M ; besides, T^-^F=Um"T^Fey^-^(W); hence, we have (3. i) . 143 144 J.-CL. TOUGERO N (3.2) We have Y(M)c¥(M). Let FeT^M) and let X' be a compact subset of ^. Let X be a compact neighborhood ofX' in 0^ and let us put Y=0(X) and <I)Q==<I)[X. Finally let s be a number >o and pi be a positive integer. We have only to prove the following result: (3.3) There exist geS(Sly) with g=i in a neighborhood of Y, and GeM, such that: |(D^)F-Y(G)l^<c. This easily results from two lemmas. We first give a definition: Definition (3.4). — A subset K of Y is (a, m) -elementary if the following conditions are verified: 1 ) There exists a constant G>o such that, V.yeX and V^eK: \W-y\^G.d[x,<S>-\y)Y. 2) The dimension of the real vector space Y^-i^(T^M) is constant, for jyeK. Lemma (3.5). — Let us suppose that 0 verifies the condition (H) and let Z be a compact and non empty subset of Y. Then, there exists a closed set E(Z)cZ such that each compact set in Z—E(Z) is (a, m)-elementary {m is an arbitrary integer, but a is the real number associated to X and Y by the condition (H)). Proof. — With the notations of (1.1)3 the function: \3y^T[y) is lower semi- continuous (because, for a fixed x, the mapping \3y\->d{x, ^"^(j^) is lower semi- continuous) . So there exists an open dense set Zg in Z, such that this function is bounded on each compact set in ZQ. Let VeZo: if x° belongs to the fiber Oo"1^0)? we have: lim d{x°, (^(j^^o. yeZ o (Indeed, by hypothesis, there exists a constant Oo such that, for each j^eZ^ in a neighborhood ofy°, we have | f —y \ > C. d(x°, O"1^))01. Let X(Y) ={x\f), .... x^f)} be an m-essential subset of the fiber ^(f) for feZ^ We can associate to each j^eZo a subset X(j^) ={x1[y), . . ., x^y)^ of ^o"1^), so that lim x\y) = x\y°) for i==i, .. ., s. Clearly, we have the following inequalities, for | y—y^ \ small enough: dim^^M) > dim^^M) ^ dim^^(T^M)-dim^^^(T^M). So the function ZoBji-^dimaT^i^^T^M) is lower semi-continuous, bounded with integer values. Therefore, there exists an open and non empty subset Z^ of ZQ in which this function is constant. Then it suffices to put E(Z)==Z—Z^. Lemma (3.6). — Let K. be a compact and (a, m)-elementary subset ofY, and let us suppose that m^[LOL. Then we can find ^e<?(Dp) with g==i in a neighborhood of K, and GeM, such that: |0^)F-Y(G)|^<c. 144 AN EXTENSION OF WHITNEY'S SPECTRAL THEOREM 145 Proof. — The following proof takes inspiration from the proof of the spectral theorem (B. Malgrange [i], lemma (1.4), chap. II). Let j/°eK. By hypothesis, there exists a neighborhood Vyo of y° and G^, . . . , Gj, in M such that for j/eV^nK, Y^^)(T^Gi), . . ., Y^(T^) is a basis of the real vector space Y^i^(T^M). Hence there exist continuous functions \,...,\ on Vyo n K, such that: T^^F^y^^^Sx^^.T^G,) t=i for all j/eVyonK. Using a partition of unity, we can find G^, . . ., G^eM, continuous functions X^, . . ., \ on K, and a constant C, such that, for all ^eK: ^o-1^)17 = ^o-^ .^ ^(^ • ^T^y and sup |\(^)[<G. 1^^ ye K /' ____ Let us put Gy== S\(j;)G,; clearly, F-Y(Gy) is T^-flat on Oo"1^)- Let ^ be a modulus of continuity on the compact set X for F, Y(Gi), ...,Y(G^) : there exists a constant Ci>o such that G^.co is a modulus of continuity on X for all functions F-Y(G,),^eK. ____ Let xeX' and ae^^y) such that d{x, Oo-1^))^^, a). The function F-Y(Gy) being m-flat at a, we have: |DfcFM-Dfcy(G,)^)|=[(R;(F~T(G,)))feM|<C,.^,Oo-l^))w-lfcl.^ Clearly, there exists a constant C[ such that 6?(^, ^o"1^))^^!-^? O"^^)) for all A:eX' and j^eK. Hence, the compact K being (a, m) -elementary and w>pia, we see that there exist a constant Gg and a modulus of continuity G)' such that: (3.6.1 ) ID^M-D^G^MI^C^IOW-^l^-'^.co'dOM-j/l) for all 72-integers A; such that |A:|<pi, all A:eX' and all j/eK. Let at be a real number >o. The open cubes of side 2d, centered at the points (j\d, . . .,jpd) (ji, . . .,jp are integers) constitute an open covering 3 ofR^ . Let^ (zeS) be a partition of unity subordinate to 3 such that, for [ k \ < (JL, (3.6.2) ^JD^OOl;^ for all yeW (Cg is a constant only depending on (x and j&). Let 3' be the finite family of those cubes L in 3 which meet K. For Le3', letj^ be a point in Ln K. Let us put: ^.IlA- aca^-G'.• 145 146 J-CL. TOUGERO N Obviously, g == i in a neighborhood of K and: |<I>^)F-Y(G)|^'.< S sup |DW&J(F-y(GJ))W| L^3 a;ex' |fc|^{ A and so, by Leibniz's formula and (3.6.1), (3.6.2): [<D*O?)F-T(G)|^G,.(^) where C^ is independent of d. Hence if we choose d sufficiently small, the lemma follows. Proof of (3.3). — First let us decompose the compact set Y with the help of Lemma (3.5). Let a be the real number associated to X and Y by the condition (H) and let m be an integer J> px. Let T be a well ordered set. We construct, by transfinite induction, a mapping T9T|->Y^ with values in the set of compact subsets ofY . If i denotes the first element of T, we put Y^==Y. Suppose the mapping is defined in the interval [i, Ti[: we put Y^ == D Y^, if T^ has no predecessor; on the other hand, if TI=T-(-I? we put: Y^IE(Y,) if Y^+0 and Y^=0 if Y^=0. If the cardinal of T is sufficiently large, there exist some T such that Y^==0. Let YI be the smallest element rof T such that Y^=0: we have v^=v+ 1 tor a- ^T (otherwise, we should have fl Y^==0, which is absurd, because the Y^, T<Vi3 are compact and non empty sets such that Y^^^cY^. for each r). Let us consider the following assertion: (HJ There exist g^ in ^(ftp) with ^==1 in a neighborhood V^ of Y^, and G^ in M, such that [O^JF-^G^I^s. The set of all T such that (H^) is true is non empty: Indeed, by (3.6), it contains v (because Y^, is a compact and (a, ^-elementary set). Let T^ be the smallest element of this set: we have to show that T^ == i. Indeed, suppose that T^>I. Necessarily, Ti=r4- i for an element reT (otherwise, we should have Y^== f1 Y^. and therefore Y^cV^, hence (H^), for a T<TI, which is absurd). We have |O*(^)F—Y(G^)|^^£'<£, with ^=i in an open neighborhood V^ ofY^. Let us put K==Y^.—V^: K is a compact and (a, m) -elementary subset of Q,p. By (3.6), applied to $*(i—^)F instead of F, there exist he^'^O.p) with h==i in a neighborhood of K, and GeM, such that: 1(I>WI~^)).F-Y(G)^<£-8/. Let us put g^g^+h-h.g^ and G^=G+G^. Clearly, ^e^(Qp), ^=i in a neighborhood of Y^, G^eM and lO'1'^) .F—^G^I^e. Hence condition (H^) is fulfilled, which is absurd. Remark (3.7). — I do not know if Theorem (1.1) is always true without the hypothesis (H): unfortunately, I have no counter-example. 146 AN EXTENSION OF WHITNEY'S SPECTRAL THEOREM 147 4. A refinement of theorem i . 2 when 0 is polynomial. Let us recall the following definition: a set in R^ is semi-algebraic if it is a finite union of subsets X^, each X^ being defined by a finite number of polynomial equalities or inequalities. The image of a semi-algebraic set by a polynomial mapping 0 : R^—^R^ is semi- algebraic (this is a fundamental result of Seidenberg and Tarski, cf. [3]), if X and Y are semi-algebraic sets in Rn and if XcY, the closure of X in Y and Y\X are semi- algebraic. Finally, it is obvious that finite unions or finite intersections of semi-algebraic sets are semi-algebraic. Let 0 be a polynomial mapping from D^ == R72 to Sly = R^ and let X and Y be compact and semi-algebraic sets in R^ and ^(R^ respectively. The following theorem improves (1.2) : Theorem (4.1). — There exists a closed and semi-algebraic set D(Y) in Y, such that Y\D(Y) is dense in Y, and constants G>o, a>o, P^o such that, for all ^eX and j^eY: (4.1.1 ) |(DM-^|>G.^O-lCy))a.^D(Y))^ Proof. — By (1.2)3 there exists an a>o (we suppose that a is an integer, which is always possible) such that, VjeY: r(y)== sup (^.o-1^/!^)--^)^. ^EXVO-^y ) Let us put D(Y)={j^EY| r is not bounded in every neighborhood of j^}. Clearly, D(Y) is closed and Y\D(Y) is dense in Y (because the mapping Y3jt->F(j^) is lower semi-continuous). Let us verify that D(Y) is semi-algebraic. First, the set A^O^.T^XxYxR^ |O(^)-^|>T.^, O-1^))'} is semi-algebraic. Indeed, A^ is the image of the semi-algebraic set A^^.^T^XxR^YxR^QK)^ and |O(^)-^|>T. l^-^l" } by the projection: XxR^YxR^XxYxR4'. Now the set A^O^eYxR^ 3x(=X such that \^{x)-jy\<,-u.d[x, O"1^))0'} is semi-algebraic, because it is the image of (XxYxR^VA^ by the projection: XxYxR + -^YxR + . Clearly, we have D(Y)x{o}=A,nYx{o}, and therefore D(Y) is semi-algebraic. Let us prove inequality (4.1.1) (the proof is similar to that of Lemma i in [4]). Let us put: Bi-^S^YxR^R^rfC^Y))^} ^={(J;^^)^|V^X,|(D(.-)-^|>T^,(]>-1^))-}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

An extension of whitney’s spectral theorem

Publications mathématiques de l'IHÉS , Volume 40 (1) – Aug 6, 2007

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Springer Journals
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Copyright © 1971 by Publications mathématiques de l’I.H.É.S
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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0073-8301
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1618-1913
DOI
10.1007/BF02684697
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Abstract

by J.-G1. TOUGERON i. Notations and results. For jye'R13 and YcR^ \y\ denotes the euclidean normof> and rf(j/,Y) the euclidean distance from y to Y. If Y is empty, we write d{y, Y) == i. Let ^ denote an open set in R^ and ^{O.p) the R-algebra of all G00 real-valued functions in Qy. When j^eQp, let ^m denote the R-algebra of Taylor expansions of order m atj/ of all elements in <^(t2p); if w<oo, ^w is isomorphic to the algebra ^p/m^4'1, where m? denotes the maximal ideal of the formal power series ring ^'=^[[^1, . . . , y ]] $ if m=+co, ^m (simply written^) (1) is isomorphic to ^ (by the generalized Borel theorem). Let T^ : S{^Y-> [y^Y denote the projection associating to each function G its Taylor expansion of order m at y. If Y is a compact set in 0. we write |G[^= = sup [D^GO^I. We provide ^(ti^with its usual structure of a Fr^chet space, !/£Y |fc|^w defined by the family of all semi-norms Gh>|G|^, where Y ranges over the set of compacts in 0,y and me'N. Let M be a submodule of ^(ftyY and let us write M={GG^(^|Vj^,3G'eM so that G-G' isflatatj/}= 11 (T^-^T.M). According to a standard result of Whitney (B. Malgrange [i]), M is the closure M of M in ^(tip)^: we propose to extend this theorem. Let <I> denote a G00 function from an open set Q^ in Rn to 0. . The mapping O defines a homomorphism of R-algebras <S>*: S(ft^)3g h>^o(pe<;?(^J. Let Y be a 0*-homomorphism from ^(^)3 to <f(0.n)\ i.e. Y is a homomorphism of abelian groups and, VGe^(Q^ and V^e<?(^): V{g.G)=(S>\g) .Y(G) . For ye^ and ^eO-1^), the mapping Y induces an R-linear mapping Y^ : [^Y -> (e^)', so that T^oY==Y^oT^. For XcO"^^), we note Y^ the R-linear mapping (^y^^Vn-(T^(V))^^x6 n (^rw)r. Finally, let T^ be the mapping ^(^n)r^F^(T^F)^x^ H (^T1)'- fl;ex 3? £ X . We propose to determine the closure Y(M) of Y(M) in ^{f^nY- Therefore, let us write ^^^{Fe^tiJ^Vj/e^, 3GeM such that Y(G)-F is flat on Y-^j)} -.ro,^-1^'1^-1^0^^- (1) We shall omit afterwards the index m, if w == + oo, and shall write: Ty,^, ... instead of T°°, T?, ... 18* 140 J.-CL. TOUGERO N We shall prove the following result: Theorem (1.1). — Let us suppose that 0 verifies the following condition: (H) For all compact sets Xc^ and Yc^Qp, there exists a constant ocj>o such that, Vj/eY: roo= sup (^<D-l(^))a/[(DW-^|)<oo. a^exvo-^y) T^ Y(M)==Y(M). It is easy to find G00 mappings 0 which do not satisfy this condition. Nevertheless, we shall prove the following result: Theorem (1.2). — An analytic mapping 0 verifies the condition (H). Both following paragraphs are devoted to the proofs of these theorems which are independent of each other. In the last paragraph, we give a refinement of the Theorem (1.2), when 0 is a polynomial mapping. 2. Proof of theorem i . 2. Definition (2.1). — Let 3 be a finitely generated ideal of a subring of the ring of germs at x° in K^ of continuous functions with real values. Let 91(^)3 . . ., <pg(^) denote real valued functions, continuous in a neighborhood of x° and such that their germs at x° generate 3. Let V(3) be the set of their zeros. We say that 3 verifies a Lojasiewic^, inequality of order a^>o (or simply that 3 verifies J2^(a)) if there exist a constant G>o and a neighborhood V of x° such that, V^eV, Sly.M^G.^V^))". i==l Let iQp be an open set in R7', ^ an open set in If, j/=(j^, ...,jp) and x=(x^y .. ., x^) coordinate systems in Q.y and ^ respectively. Let (9 be the sheaf of germs of analytic functions with real values on Q.^x0.p; ^ a sheaf of ideals, analytic and coherent on Q^xOp. For (^°, y°)e 0.^x0. „ , we denote ^yo) the stalk of ^ at the point {x°,jy°). Let 9^, . . ., cp, be generators of the ideal ^^y we denote ^,1/0) the ideal generated by (pi^,./), ...,(p,(^,j/0) in the ring O^y^ of germs at {x0,^) in Q^x{^°} of analytic functions with real values. Permuting x and y, we define similarly the ideal -^yo) of ^\y°}' Finally, let V(J^) be the set of zeros ofJ^. Theorem (1.2 ) is an easy consequence of the following one (Lojasiewicz inequality with a parameter): Theorem (2.2). — Let X be a compact set in 0.^, Y a compact set in £ip. There exists a^o such that the ideal J^ verifies J^(a), V(^,j/)eXxY. Indeed, let us suppose this theorem is true, and let 0 be an analytic mapping. Let ^ denote the analytic and coherent sheaf generated on O.^x0.p by ^iM—j^, .... O {x)—jy . Let X, Y be compact sets in ^, Q.y respectively. By (2.2) applied to J^, V(,y°,^) 6=XxY, there exists a constant G^y)>o such that for ^ in a neighborhood of 24^ AN EXTENSION OF WHITNEY'S SPECTRAL THEOREM 141 ^rIOM—j/I^G^^.^O- 1^)) 01 . Hence, the set X being compact, there exists a constant Gy>o such that, V^eX: lOW-j^C^O-1^))-. Clearly, condition (H) follows. Proof of (2.2). — Obviously, condition JSf(oc) is verified, with a==o, for (^,j/)^V(J^). The set XxY being compact, it suffices to find, for (^.j^eV^), an oc;>o such that J^y) verifies JSf(a) for (A;,J^) in a neighborhood of (A;0,^0). We shall suppose that {x°,y°) is the origin of B^xR^. Now, it is enough to prove the following result: (2.3) There exists an a^o such that ^^ verifies JSf(oc) for (o^eV^) and \y\ small enough. Indeed, let 9i(^), . . ., <pg(A:,^) generate ^ in a neighborhood of (o, o), and let us consider the sheaf ^ generated on a neighborhood of the origin of R^xB^xR^ by 9i(^+^), . . ., 9,(x+^,^). By (2.3) applied to the sheaf / (with the para- meter {^y} instead ofj/), there exists an ocj>o such that ^^=J^^ verifies .JSf(oc) for [^y ) in a neighborhood of the origin. Proof of (2.3). — We proceed by induction on the height k of the ideal A) Q) . There exist sheafs of ideals ^>1, . . ., ^r, analytic coherent on a neighborhood of the origin of R^R^, such that <^o)? • • "> %o) are prime ideals of height J>/;, and an integer P^i, such that: ^D(^ln...n^r)p. Clearly, if ^j^ verifies J§f(a,) for y small enough, e^o ^ verifies J§f(p S a,) forj/ small enough. Hence, we may suppose that ^[0,0} ls prime and its height equals k. Let <p(j/) be analytic in a neighborhood of the origin of oxR^ and null in V(J^) n (o X R^) in a neighborhood of the origin. Let / be the analytic coherent sheaf on a neighborhood of the origin of R^R^, generated by ^ and 9: obviously, ^,y}==^,y] forj^ small enough. If cp(^J^, we get ht /^Qy>k and hence the result is proved by the induction hypothesis. Therefore, we may suppose that cp(=eXo 0)5 i-e. ^o)3-^) a^ -%o) ls the ^eal of germs <p(j;) null in V(A) n (oxR^) . Lemma (2.4). — 14Wz ^ preceding hypothesis, let k—l be the height of the prime ideal -%o)- After an eventual permutation on the coordinates x^, ...,^, there exist <pi, ... , 9^^0,0) such that ^= yl? " > ? ?^ ^0,0)' -u ^1 5 • • • 5 ^J Proo/'. — We proceed by induction on the height k of e^o ^. Let us suppose that k>{. There is a sequence (o,Y)eV(J^), V-^o, such that for each i: ^ .4=0 (otherwise ^^ would be generated by ^,o))- After an eventual linear change of coordinates on the variables ^, ...,A^ , we know (following the analytic preparation theorem, Malgrange [i]) that there exists, for each i, a distinguished polynomial T^=^i+^,^(^J;).^^~l+...+^.,^(^,J;)£^o,^ (we write x'=(x^...,x ^ and the a^ 141 I^ J.-CL. TOUGERO N are analytic functions of {x',y) in a neighborhood of (o,y)). Besides, we may suppose that ~»~^^W (Ir^cd, there exists a smaller integer B,>o such that ^_^4-J •• 1 a3,w ftc3'4'1 ' t0-!'1)' we have only to substitute ——— for Y,.) Hence, there exists y^ „, such that 8<pl d- <r 1 ^ (oto)- Let Q' be the sheaf of germs of analytic functions with real values on R^xR^^jiQ eR-xR^l^o } and let us write ^'=^n 0'' . There exists an integer ^ such that htJ^.)=A: for i^g; besides, <P((», ,,) | ^o,».) is a finitely generated module over ^(o^Ko.i,.) and hence their Krull dimensions are equal (by the Gohen-Seidenberg theorem, Malgrange [i]; th. (5.3), chap. Ill); therefore ht^^=k-i for ^. Since ^ is prime, ht ^ o) = ht J^,, for (• large enough, so that: ht^=k-r, finally, J^op.-%o)- Applying the induction hypothesis to the sheaf Jf '(after an eventual permutation on the variables ^, . .. , A-J, we see that there exist <pa, .. ., y^e^ o, such that ^""J^^o)- Hence: 1^^2, . . ., A"/J P^ ...,y^_8cp i D(<pa, ...,y^) ^ ^ D(^,...,^ ) ^•D(^,...,^)^(0.0'- Since ht^<,)=/:-/ and J^, is prime, there exist y/^, ...,y,e^^, such that, after an eventual permutation on the coordinates Vi, . .. y : ,_P(<P^l, ••-,9, ) . D(yi,...,y,) s2 D(^,...^_/)^o,o>, hence -D(^ .. ., ./,^ .. ..^) = ^•^0,0). By the jacobian criterion for regular points, the localized ring (^g g,) is regular of dimension k and its maximal ideal is generated by y,, .. ., ^. Hence"t'here exists S3^(o,oMo,o) such that: ^3-^0,0)^^1, ••-,%) . Let S be analytic in a neighborhood of (o, o)€R"+p and inducing the germ ^.^3 at the origin. Let / be the sheaf of ideals generated, on a neighborhood of the origin of R^, by ^ and ^. For (o,y) eV(^), y small enough: — There exists a>o such that ^o%)=^o",)+^.%,) verifies ^(oc) (because nt ^0,0) ^t^o.o) ''•"d we apply the induction hypothesis). — ^•^,y) is contained in the sub-ideal ofJ%y, generated by y^, . .. , ^. — Finally, ^ belongs to the ideal generated in % .1 by the jacobian D^1' ••• > ^ ) D(^, ...,^ ) • So Theorem (2.3) is an immediate consequence of the following lemma (Tougeron and Merrien [2], prop. 3, chap. II): Lemma (2.5). — Let 3 be a finitely generated ideal of the ring ^ of germs at the origin in R" of 0°° functions with real values. Let y,, . . ., y/e3 and ^ belonging to the ideal generated 142 AN EXTENSION OF WHITNEY'S SPECTRAL THEOREM 143 in < by (pi, . . ., ^ and all the jacobians ^1? '"? ^ , jo that ^. 3 C (<pi, . . ., (p^). 77^ -U(A^, . . ., ^) ^ 3^3+S.^ verifies JSf(oc), ^ zY^ 3 verifies oSf(sup(2oc, a+i)). J?^rA; (2.6). — Let 0 =(d>i , . .. , Op) be a G00 mapping from ^ to Qy. Let S be the sheaf of G00 functions with real values on ^ (or Q^,); let ^ be the sheaf of ideals D($i $ ) generated on ^ by all the jacobians _ , ' ' ' ?—^-: the set VfJQ of zeros of ^ is the D(^i, ...,^) set of singular points of the mapping 0. Let us consider the following condition: (H') V^eV(^), <^/^ is {by 0) a module of finite type over the ring €y [we set y==<S)(x)), i.e. by the Malgrange preparation theorem (Malgrange [i]): (C/^^W^-^/^+m,.^) is a real vector space of finite dimension (nty: maximal ideal of S). The condition (H') is a very strong one; nevertheless, it is a generic one, i.e. it is verified on an open dense subset of the space of G00 mappings from Q^ to Q. , this space being provided with the Whitney topology. Besides, (H') implies (H). Indeed, let X and Y be compact sets in ^ and ^ respectively. By hypo- thesis, there exists an a>o such that, V^j^eXxY, the ideal generated by Df O $ ) °iW-J^ • • •. ^pW-Vp and all the jacobians _ 1? " > ? p/ in S^^ (ring of germs U(^3 . . ., X^y) at {x°,y°) in R^V} ofG°° functions with real values), verifies J§f(a). By Lemma (2.5), the ideal generated by OiW—j^, . . ., O^)—^, in <^yo) verifies ^(a'), with an a' independent of the point (A^j/^eXxY. Clearly, the condition (H) follows. 3. Proof of theorem 1.1. With the notations of § i, we must show that: ^(M^^M). (3.1) We have: T(M)cY(M). Let FeT(M) and let ye0.y. A finite subset X^ of O"^^) will be called m-essential (m is a positive integer), if ker Y^-i^ = ker T^; clearly, there always exist w-essential sets X^ such that card X^< card {^Y. Let X be a finite subset of (S)~l{y) containing such an X^. By hypothesis, T^F is in the closure of the finite dimensional real space Y^(T^M), and therefore belongs to it. So there exist G^ with G^eT^M such that: T^F==y^(G^) and T^F=Y^(GW). Obviously, G^-G^eker Y^=ker Y^$ thus: T^F^Y^G^), and X being arbitrary : T^-i^F^T^-i^G^. So, WW=(y^-x^)-l(T^-x(y)F)nT^M is a finite dimensional and non empty affine space. The inverse limit W^imW^ is then non empty and contained in UmT^M=T^M ; besides, T^-^F=Um"T^Fey^-^(W); hence, we have (3. i) . 143 144 J.-CL. TOUGERO N (3.2) We have Y(M)c¥(M). Let FeT^M) and let X' be a compact subset of ^. Let X be a compact neighborhood ofX' in 0^ and let us put Y=0(X) and <I)Q==<I)[X. Finally let s be a number >o and pi be a positive integer. We have only to prove the following result: (3.3) There exist geS(Sly) with g=i in a neighborhood of Y, and GeM, such that: |(D^)F-Y(G)l^<c. This easily results from two lemmas. We first give a definition: Definition (3.4). — A subset K of Y is (a, m) -elementary if the following conditions are verified: 1 ) There exists a constant G>o such that, V.yeX and V^eK: \W-y\^G.d[x,<S>-\y)Y. 2) The dimension of the real vector space Y^-i^(T^M) is constant, for jyeK. Lemma (3.5). — Let us suppose that 0 verifies the condition (H) and let Z be a compact and non empty subset of Y. Then, there exists a closed set E(Z)cZ such that each compact set in Z—E(Z) is (a, m)-elementary {m is an arbitrary integer, but a is the real number associated to X and Y by the condition (H)). Proof. — With the notations of (1.1)3 the function: \3y^T[y) is lower semi- continuous (because, for a fixed x, the mapping \3y\->d{x, ^"^(j^) is lower semi- continuous) . So there exists an open dense set Zg in Z, such that this function is bounded on each compact set in ZQ. Let VeZo: if x° belongs to the fiber Oo"1^0)? we have: lim d{x°, (^(j^^o. yeZ o (Indeed, by hypothesis, there exists a constant Oo such that, for each j^eZ^ in a neighborhood ofy°, we have | f —y \ > C. d(x°, O"1^))01. Let X(Y) ={x\f), .... x^f)} be an m-essential subset of the fiber ^(f) for feZ^ We can associate to each j^eZo a subset X(j^) ={x1[y), . . ., x^y)^ of ^o"1^), so that lim x\y) = x\y°) for i==i, .. ., s. Clearly, we have the following inequalities, for | y—y^ \ small enough: dim^^M) > dim^^M) ^ dim^^(T^M)-dim^^^(T^M). So the function ZoBji-^dimaT^i^^T^M) is lower semi-continuous, bounded with integer values. Therefore, there exists an open and non empty subset Z^ of ZQ in which this function is constant. Then it suffices to put E(Z)==Z—Z^. Lemma (3.6). — Let K. be a compact and (a, m)-elementary subset ofY, and let us suppose that m^[LOL. Then we can find ^e<?(Dp) with g==i in a neighborhood of K, and GeM, such that: |0^)F-Y(G)|^<c. 144 AN EXTENSION OF WHITNEY'S SPECTRAL THEOREM 145 Proof. — The following proof takes inspiration from the proof of the spectral theorem (B. Malgrange [i], lemma (1.4), chap. II). Let j/°eK. By hypothesis, there exists a neighborhood Vyo of y° and G^, . . . , Gj, in M such that for j/eV^nK, Y^^)(T^Gi), . . ., Y^(T^) is a basis of the real vector space Y^i^(T^M). Hence there exist continuous functions \,...,\ on Vyo n K, such that: T^^F^y^^^Sx^^.T^G,) t=i for all j/eVyonK. Using a partition of unity, we can find G^, . . ., G^eM, continuous functions X^, . . ., \ on K, and a constant C, such that, for all ^eK: ^o-1^)17 = ^o-^ .^ ^(^ • ^T^y and sup |\(^)[<G. 1^^ ye K /' ____ Let us put Gy== S\(j;)G,; clearly, F-Y(Gy) is T^-flat on Oo"1^)- Let ^ be a modulus of continuity on the compact set X for F, Y(Gi), ...,Y(G^) : there exists a constant Ci>o such that G^.co is a modulus of continuity on X for all functions F-Y(G,),^eK. ____ Let xeX' and ae^^y) such that d{x, Oo-1^))^^, a). The function F-Y(Gy) being m-flat at a, we have: |DfcFM-Dfcy(G,)^)|=[(R;(F~T(G,)))feM|<C,.^,Oo-l^))w-lfcl.^ Clearly, there exists a constant C[ such that 6?(^, ^o"1^))^^!-^? O"^^)) for all A:eX' and j^eK. Hence, the compact K being (a, m) -elementary and w>pia, we see that there exist a constant Gg and a modulus of continuity G)' such that: (3.6.1 ) ID^M-D^G^MI^C^IOW-^l^-'^.co'dOM-j/l) for all 72-integers A; such that |A:|<pi, all A:eX' and all j/eK. Let at be a real number >o. The open cubes of side 2d, centered at the points (j\d, . . .,jpd) (ji, . . .,jp are integers) constitute an open covering 3 ofR^ . Let^ (zeS) be a partition of unity subordinate to 3 such that, for [ k \ < (JL, (3.6.2) ^JD^OOl;^ for all yeW (Cg is a constant only depending on (x and j&). Let 3' be the finite family of those cubes L in 3 which meet K. For Le3', letj^ be a point in Ln K. Let us put: ^.IlA- aca^-G'.• 145 146 J-CL. TOUGERO N Obviously, g == i in a neighborhood of K and: |<I>^)F-Y(G)|^'.< S sup |DW&J(F-y(GJ))W| L^3 a;ex' |fc|^{ A and so, by Leibniz's formula and (3.6.1), (3.6.2): [<D*O?)F-T(G)|^G,.(^) where C^ is independent of d. Hence if we choose d sufficiently small, the lemma follows. Proof of (3.3). — First let us decompose the compact set Y with the help of Lemma (3.5). Let a be the real number associated to X and Y by the condition (H) and let m be an integer J> px. Let T be a well ordered set. We construct, by transfinite induction, a mapping T9T|->Y^ with values in the set of compact subsets ofY . If i denotes the first element of T, we put Y^==Y. Suppose the mapping is defined in the interval [i, Ti[: we put Y^ == D Y^, if T^ has no predecessor; on the other hand, if TI=T-(-I? we put: Y^IE(Y,) if Y^+0 and Y^=0 if Y^=0. If the cardinal of T is sufficiently large, there exist some T such that Y^==0. Let YI be the smallest element rof T such that Y^=0: we have v^=v+ 1 tor a- ^T (otherwise, we should have fl Y^==0, which is absurd, because the Y^, T<Vi3 are compact and non empty sets such that Y^^^cY^. for each r). Let us consider the following assertion: (HJ There exist g^ in ^(ftp) with ^==1 in a neighborhood V^ of Y^, and G^ in M, such that [O^JF-^G^I^s. The set of all T such that (H^) is true is non empty: Indeed, by (3.6), it contains v (because Y^, is a compact and (a, ^-elementary set). Let T^ be the smallest element of this set: we have to show that T^ == i. Indeed, suppose that T^>I. Necessarily, Ti=r4- i for an element reT (otherwise, we should have Y^== f1 Y^. and therefore Y^cV^, hence (H^), for a T<TI, which is absurd). We have |O*(^)F—Y(G^)|^^£'<£, with ^=i in an open neighborhood V^ ofY^. Let us put K==Y^.—V^: K is a compact and (a, m) -elementary subset of Q,p. By (3.6), applied to $*(i—^)F instead of F, there exist he^'^O.p) with h==i in a neighborhood of K, and GeM, such that: 1(I>WI~^)).F-Y(G)^<£-8/. Let us put g^g^+h-h.g^ and G^=G+G^. Clearly, ^e^(Qp), ^=i in a neighborhood of Y^, G^eM and lO'1'^) .F—^G^I^e. Hence condition (H^) is fulfilled, which is absurd. Remark (3.7). — I do not know if Theorem (1.1) is always true without the hypothesis (H): unfortunately, I have no counter-example. 146 AN EXTENSION OF WHITNEY'S SPECTRAL THEOREM 147 4. A refinement of theorem i . 2 when 0 is polynomial. Let us recall the following definition: a set in R^ is semi-algebraic if it is a finite union of subsets X^, each X^ being defined by a finite number of polynomial equalities or inequalities. The image of a semi-algebraic set by a polynomial mapping 0 : R^—^R^ is semi- algebraic (this is a fundamental result of Seidenberg and Tarski, cf. [3]), if X and Y are semi-algebraic sets in Rn and if XcY, the closure of X in Y and Y\X are semi- algebraic. Finally, it is obvious that finite unions or finite intersections of semi-algebraic sets are semi-algebraic. Let 0 be a polynomial mapping from D^ == R72 to Sly = R^ and let X and Y be compact and semi-algebraic sets in R^ and ^(R^ respectively. The following theorem improves (1.2) : Theorem (4.1). — There exists a closed and semi-algebraic set D(Y) in Y, such that Y\D(Y) is dense in Y, and constants G>o, a>o, P^o such that, for all ^eX and j^eY: (4.1.1 ) |(DM-^|>G.^O-lCy))a.^D(Y))^ Proof. — By (1.2)3 there exists an a>o (we suppose that a is an integer, which is always possible) such that, VjeY: r(y)== sup (^.o-1^/!^)--^)^. ^EXVO-^y ) Let us put D(Y)={j^EY| r is not bounded in every neighborhood of j^}. Clearly, D(Y) is closed and Y\D(Y) is dense in Y (because the mapping Y3jt->F(j^) is lower semi-continuous). Let us verify that D(Y) is semi-algebraic. First, the set A^O^.T^XxYxR^ |O(^)-^|>T.^, O-1^))'} is semi-algebraic. Indeed, A^ is the image of the semi-algebraic set A^^.^T^XxR^YxR^QK)^ and |O(^)-^|>T. l^-^l" } by the projection: XxR^YxR^XxYxR4'. Now the set A^O^eYxR^ 3x(=X such that \^{x)-jy\<,-u.d[x, O"1^))0'} is semi-algebraic, because it is the image of (XxYxR^VA^ by the projection: XxYxR + -^YxR + . Clearly, we have D(Y)x{o}=A,nYx{o}, and therefore D(Y) is semi-algebraic. Let us prove inequality (4.1.1) (the proof is similar to that of Lemma i in [4]). Let us put: Bi-^S^YxR^R^rfC^Y))^} ^={(J;^^)^|V^X,|(D(.-)-^|>T^,(]>-1^))-}.

Journal

Publications mathématiques de l'IHÉSSpringer Journals

Published: Aug 6, 2007

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