C^m Semialgebraic Sections Over the Plane
aa r X i v : . [ m a t h . C A ] J a n C m Semialgebraic Sections Over the Plane
Charles Fefferman, Garving K. LuliFebruary 2, 2021
In this paper we settle the two-dimensional case of a conjecture involving unknown semial-gebraic functions with specified smoothness.Recall that a semialgebraic set E ⊂ R n is a union of finitely many sets of the form { x ∈ R n : P ( x ) , P ( x ) , · · · , P r ( x ) > , and Q ( x ) = Q ( x ) = · · · = Q s ( x ) = 0 } for polynomials P , · · · , P r , Q , · · · , Q s on R n . (We allow the cases r = 0 or s = 0.)A semialgebraic function φ : E → R D is a function whose graph { ( x, φ ( x )) : x ∈ E } is asemialgebraic set.We define smoothness in terms of C m and C mloc . Here, C m (cid:0) R n , R D (cid:1) denotes the space ofall R D -valued functions on R n whose derivatives up to order m are continuous and boundedon R n . C mloc (cid:0) R n , R D (cid:1) denotes the space of R D -valued functions on R n with continuousderivatives up to order m . If D = 1, we write C m ( R n ) and C mloc ( R n ) in place of C m (cid:0) R n , R D (cid:1) and C mloc (cid:0) R n , R D (cid:1) , respectively.To motivate our conjecture, we pose the following problems. Problem 1 (Semialgebraic Whitney Problem; see [43].)
Fix m ≥ . Let φ : E → R be semialgebraic. Suppose φ extends to a C mloc function on R n . Does it necessarily extend toa C mloc semialgebraic function on R n ? Problem 2 (Linear Equations)
Fix m ≥ . Consider the linear equation (1) A F + · · · + A D F D = f for unknowns F , · · · , F D on R n , where A , · · · , A D , f are given semialgebraic functions. Ifequation (1) admits a C mloc solution F , · · · , F D , does it necessarily admit a C mloc semialgebraicsolution? More generally, in place of (1) we can consider underdetermined systems of linear equa-tions.Problem 1 was raised by Bierstone and Milman in [43].Note that m is fixed in the above problems so we are not allowed to lose derivatives.1roblems 1 and 2 are instances of a more general question. The purpose of this paper isto settle that question, and in particular provide affirmative answers to Problems 1 and 2,in the case of C mloc ( R ).To pose our more general question, we set up notations and give a few basic definitions.Fix m ≥
0. If F ∈ C mloc ( R n ) and x ∈ R n , we write J x ( F ) (the “jet” of F at x ) to denotethe m -th degree Taylor polynomial of F at x .Thus, J x ( F ) belongs to P , the vector space of all such polynomials.For x ∈ R n , P, Q ∈ P , we define P ⊙ x Q = J x ( P Q ). The multiplication ⊙ x makes P intoa ring, denoted by R x , the “ring of m -jets at x ”. We have J x ( F G ) = J x ( F ) ⊙ x J x ( G ) for F, G ∈ C mloc ( R n ).We consider vector-valued functions F = ( F , · · · , F D ) : R n → R D , and we write F ∈ C mloc (cid:0) R n , R D (cid:1) if each F i ∈ C mloc ( R n ). We define J x F = ( J x F , · · · , J x F D ) ∈ P ⊕ · · · ⊕ P .Under the natural multiplication Q ⊙ x ( P , · · · , P D ) := ( Q ⊙ x P , · · · , Q ⊙ x P D ) ,the vector space P ⊕ · · · ⊕ P becomes an R x module, which we denote by R Dx .We will discuss R x -submodules of R Dx ; we allow both { } and R Dx as submodules of R Dx .Fix m, n, D , and a subset E ⊂ R n . For each x ∈ E , let H ( x ) = f ( x ) + I ( x ) ⊂ R Dx be given, where f ( x ) ∈ R Dx and I ( x ) ⊂ R Dx is an R x -submodule. Then the family(2) H = ( H ( x )) x ∈ E is called a “bundle” over E . H ( x ) is called the fiber of H at x . Remark 1.1
We remark that our notion of bundle differs from the notion of a bundleconsidered previously (e.g, [28]). In the present version, we do not require E to be compactand we require all the fibers H ( x ) to be non-empty. When m, n, D are not clear from context, we speak of a “bundle with respect to C mloc (cid:0) R n , R D (cid:1) ”.If H is given by (2) and E ′ ⊂ E , then we write H| E ′ to denote the bundle ( H ( x )) x ∈ E ′ ,and refer to H| E ′ as the restriction of H to E ′ .A “section” of the bundle H in (2) is a vector-valued function F ∈ C mloc ( R n , R D ) suchthat J x F ∈ H ( x ) for all x ∈ E .Note that sections F belong to C mloc (cid:0) R n , R D (cid:1) by definition.The bundle (2) is called “semialgebraic” if { ( x, P , · · · , P D ) : R n ⊕ P ⊕ · · · ⊕ P : x ∈ E, ( P , · · · , P D ) ∈ H ( x ) } is a semialgebraic set.We can now state our general problem. Problem 3
Let H = ( H ( x )) x ∈ E be a semialgebraic bundle with respect to C mloc (cid:0) R n , R D (cid:1) . If H has a section, does it necessarily have a semialgebraic section? H must belong to C mloc for fixed m , so we are not allowedto lose derivatives.One checks easily that Problems 1 and 2 are instances of Problem 3.Indeed, suppose φ : E → R is semialgebraic, as in Problem 1. Set H = ( H ( x )) x ∈ E , where H ( x ) = { P ∈ P : P ( x ) = φ ( x ) } .Then H is a semialgebraic bundle, and a section of H is precisely a function F ∈ C mloc ( R n )such that F = φ on E .Similarly, given an equation (1) as in Problem 2, set H = ( H ( x )) x ∈ R n with H ( x ) = (cid:8) ( P , · · · , P D ) ∈ P D : A ( x ) P ( x ) + · · · + A D ( x ) P D ( x ) = f ( x ) (cid:9) .Then H is a semialgebraic bundle, and a section of H is precisely a solution F = ( F , · · · , F D ) ∈ C mloc (cid:0) R n , R D (cid:1) of equation (1).In this paper, we settle the two-dimensional case of Problem 3. Theorem 1
Let H be a semialgebraic bundle with respect to C mloc (cid:0) R , R D (cid:1) . If H has asection, then it has a semialgebraic section. We give a quick sketch of the proof of Theorem 1.By a change of coordinates and a partition of unity, we may localize the problem to asmall thin wedge Γ( c ) = { ( x , x ) ∈ R : x ∈ [0 , c ] , ≤ x ≤ x } . More precisely, it is enough to prove that H| Γ( c ′ ) has a section for sufficiently small c ′ .We may assume also that our bundle H = ( H ( x , x )) ( x ,x ) ∈ Γ( c ) satisfies H ((0 , { } .We analyze what it means for a given F = ( F , · · · , F D ) ∈ C mloc (cid:0) R n , R D (cid:1) with J (0 , F = 0to be a section of H . Our analysis produces finitely many semialgebraic curves γ , γ , · · · , γ s max in Γ ( c ), and we find that F is a section of H if and only if • F ( x , x ) and its x -derivatives up to order m satisfy finitely many linear equations onthe γ s and • F satisfies finitely many linear equations on Γ( c ) \ ( γ ∪ · · · ∪ γ s max ) . The curves γ s have the form γ s = { ( x, ψ s ( x )) : x ∈ [0 , c ] } for semialgebraic functions ψ , · · · , ψ s max of one variable.The heart of our proof is to use the above characterization to produce finitely many linearequations and inequalities for unknown functions ξ lsk ( x ) of one variable ( l = 0 , · · · , m ; k =1 , · · · , D ; s = 1 , · · · , s max ) with the following properties: (A) If F = ( F , · · · , F D ) ∈ C mloc (cid:0) R , R D (cid:1) is a section of H then the functions(3) ξ lsk ( x ) = ∂ lx F k ( x , x ) (cid:12)(cid:12) x = ψ s ( x ) satisfy the above equations and inequalities for x ∈ [0 , c ]; and conversely3 B) If semialgebraic functions ξ lsk ( x ) satisfy the above equations and inequalities for x ∈ [0 , c ], then for some small c ′ < c there exists a semialgebraic section F = ( F , · · · , F D )of H| Γ( c ′ ) such that (3) holds for x ∈ [0 , c ′ ].We can easily deduce Theorem 1 from (A) and (B), as follows.Because H| Γ( c ) has a section, (A) tells us that the relevant equations and inequalities forthe ξ lsk admit a solution.Because all functions appearing in those equations and inequalities are semialgebraic(except perhaps the unknowns ξ lsk ), it follows easily that we may take the ξ lsk ( x ) to de-pend semialgebraically on x . Thanks to (B), we obtain a semialgebraic section of H| Γ( c ′ ) ,completing the proof of Theorem 1. See Section 7 for details.Let us recall some of the literature regarding Problems 1, 2, 3. The literature on Whit-ney’s extension problem goes back to the seminal works of H. Whitney [41, 42], and includesfundamental contributions by G. Glaeser [31], Yu. Brudnyi and P. Shvartsman [8–11], E.Bierstone, P. Milman, and W. Paw lucki [3–5], as well as our own papers [13–26]. In the semi-algebraic (and o -minimal) setting , the analogue of the classical Whitney extension theoremis due to K. Kurdyka and W. Paw lucki [34] and A. Thamrongthanyalak [39].Problem 1 in the setting of C loc ( R n ) was settled affirmatively by M. Aschenbrenner andA. Thamrongthanyalak [1]. Our results on Problem 3 imply an affirmative solution for C mloc ( R ). For C mloc ( R n ) with m ≥ n ≥
3, Problems 1, 2, 3 remain open.The problem of deciding whether a (possibly underdetermined) system of linear equationsof the form (1) admits a C loc solution was proposed by Brenner [7], and Epstein-Hochster [12].Two independent solutions to this problem appear in Fefferman-Koll´ar [27]. Fefferman-Luli [30] solved the analogous problem for C mloc ( m ≥ C loc -rational functions, even though A , · · · , A D and f are polynomials and a C loc solution ( F , · · · , F D ) exists. They showed that x x f + ( x − (1 + x ) x ) f = x has acontinuous semialgebraic solution but no continuous rational solution ( f , f ) ∈ C loc ( R , R ).However, [40] shows that a semialgebraic C loc solution exists, and [33] shows that a solutionby C loc semialgebraic functions exists for Problems 1 and 2 posed over R , again provided A , · · · , A D , f are polynomials.A recent paper of Bierstone-Campesato-Milman [2] shows that given a system of equations(1) with semialgebraic data A i , f , there exists a function r : N → N independent of f suchthat if the system (1) admits a C r ( m ) loc solution, then it admits a semialgebraic C mloc solution.The result of Bierstone-Campesato-Milman is more general than the version stated above;it applies to suitable o -minimal structures. Acknowledgement.
We are grateful to Matthias Aschenbrenner, Edward Bierstone,Jean-Baptiste Campesato, Fushuai (Black) Jiang, Bo’az Klartag, J´anos Koll´ar, Pierre Mil-man, Assaf Naor, Kevin O’Neill, Wies law Paw lucki, and Pavel Shvartsman for their interestand valuable comments. We would also like to thank the participants of the 11-th Whitneyworkshop for their interest in our work, and we thank Trinity College Dublin, for hostingthe workshop. The first author is supported by the Air Force Office of Scientific Research(AFOSR), under award FA9550-18-1-0069, the National Science Foundation (NSF), under4rant DMS-1700180, and the US-Israel Binational Science Foundation (BSF), under grant2014055. The second author is supported by NSF Grant DMS-1554733 and the UC DavisChancellor’s Fellowship.
A function f : R n → R is called a Nash function if it is real-analytic and semialgebraic.Write B ( x, r ) to denote the ball of radius r about x in R n .The dimension of a semialgebraic set E ⊂ R n is the maximum of the dimensions of allthe imbedded (not necessarily compact) submanifolds of R n that are contained in E .We recall a few definitions from the Introduction.Fix m, n, D , and a subset E ⊂ R n . For each x ∈ E , let(4) H ( x ) = f ( x ) + I ( x ) ⊂ R Dx be given, where f ( x ) ∈ R Dx and I ( x ) ⊂ R Dx is an R x -submodule. Then the family H = ( H ( x )) x ∈ E is called a bundle over E . H ( x ) is called the fiber of H at x .When m, n, D are not clear from context, we speak of a “bundle with respect to C mloc (cid:0) R n , R D (cid:1) ”.If H is given by (4) and E ′ ⊂ E , then we write H| E ′ to denote the bundle ( H ( x )) x ∈ E ′ ,and refer to it as the restriction of H to E ′ . If H = ( H ( x )) x ∈ E and H ′ = ( H ′ ( x )) x ∈ E arebundles, H ′ is called a subbundle of H if H ′ ( x ) ⊂ H ( x ) for all x ∈ E . We write H ⊃ H ′ todenote that H ′ is a subbundle of H .What we called a “bundle” in [28] we now call a “classical bundle”.The definition is as follows. Fix m, n, D . Let E ⊂ R n be compact. A classical bundle over E is a family H = ( H ( x )) x ∈ E of (possibly empty) affine subspaces H ( x ) ⊂ P D , parametrizedby the points x ∈ E , such that each non-empty H ( x ) has the form H ( x ) = ~P x + ~I ( x )for some ~P x ∈ P D and some R x -submodule ~I ( x ) of P D .When m, n, D are not clear from context, we speak of a “classical bundle with respect to C m ( R n , R D )”.We remark again that our notion of bundle differs from the notion of bundles consideredpreviously (e.g., [28]). In the present version, we do not require that E be compact and werequire all the fibers H ( x ) to be non-empty.A section of the bundle H is a vector-valued function F ∈ C mloc ( R n , R D ) such that J x F ∈ H ( x ) for all x ∈ E . A section of a classical bundle H is a vector-valued function F ∈ C m ( R n , R D ) such that J x F ∈ H ( x ) for all x ∈ E .5 Tools
Given a bundle H = ( H ( x )) x ∈ E for C mloc ( R n , R D ) or a classical bundle H = ( H ( x )) x ∈ E for C m ( R n , R D ), we define the Glaeser refinement H ′ = ( H ′ ( x )) x ∈ E as follows: (GR) Let x ∈ E . A given P ∈ H ( x ) belongs to H ′ ( x ) if and only if the followingholds. Given ǫ >
0, there exists δ > x , · · · , x k ∈ B ( x , δ ) ∩ E , where k is a large enough constant depending only on m , n , and D , there exist P i ∈ H ( x i )( i = 1 , · · · , k ), such that | ∂ α ( P i − P j )( x i ) | ≤ ǫ | x i − x j | m −| α | , for all | α | ≤ m, ≤ i, j ≤ k .A bundle or a classical bundle H is Glaeser stable if H ′ = H .Note that the Glaeser refinement H ′ of H may have empty fibers, even if H has none.In that case, we know that H has no sections. If H is a classical bundle, then so is H ′ . If H is a bundle and no fibers of H ′ are empty, then H ′ is a bundle. Both for bundles andfor classical bundles, every section of H is a section of H ′ . (See [28] for the case of classicalbundles; the elementary proofs carry over unchanged for bundles.) Note in particular thatif a given bundle H has a section, then H ′ has no empty fibers, hence H ′ is a bundle and H ′ has a section.Starting from a classical bundle H , or a bundle H with a section, we can perform iteratedGlaeser refinement to pass to ever smaller subbundles H (1) , H (2) , etc., without losing sections.We set H (0) = H , and for l ≥
0, we set H ( l +1) = (cid:0) H ( l ) (cid:1) ′ . Thus, by an obvious induction on l , we have H = H (0) ⊃ H (1) ⊃ · · · , yet H and H ( l ) have the same sections for all l ≥ H = ( H ( x )) x ∈ E is a semialgebraic bundle with respect to C mloc ( R n , R D ), by an obviousinduction on l , we have H ( l ) ( x ) depends semialgebraically on x , where H ( l ) = ( H ( l ) ( x )) x ∈ E . In principle, each H ( l ) can be computed from H . We remark that iterated Glaeser refine-ment stabilizes after finitely many iterations (i.e. for a large enough integer l ∗ determined by m, n, D , we have H ( l ∗ +1) = H ( l ∗ ) ; thus H ( l ∗ ) is Glaeser stable. See [28] for the case of classicalbundles; the argument, which goes back to Glaeser [31] and Bierstone-Milman-Paw lucki [4,5],applies unchanged for bundles. We call H ( l ∗ ) the stable Glaeser refinement of H .)The main results of [28] give the following Theorem 2
For a large enough integer constant l ∗ determined by m, n, and D , the followingholds. Let H be a classical bundle with respect to C m (cid:0) R n , R D (cid:1) . Let H (0) , H (1) , H (2) , · · · beits iterated Glaeser refinements. Then H has a section if and only if H ( l ∗ ) has no emptyfibers. Suppose H ( l ∗ ) has no empty fibers. Let x ∈ E and let P belong to the fiber of H ( l ∗ ) at x . Then there exists a section F of the bundle H , such that J x ( F ) = P . Moreover, thereexists a constant k depending only on m, n, and D such that the following holds: Suppose H = ( H ( x )) x ∈ E is a Glaeser stable classical bundle. Assume the following holds for someconstant M > : Given x , · · · x k ∈ E , there exist polynomials P , · · · , P k ∈ P D , with P i ∈ H ( x i ) for ≤ i ≤ k ; | ∂ α P i ( x i ) | ≤ M for all | α | ≤ m, ≤ i ≤ k ; and | ∂ α ( P i − P j )( x j ) | ≤ M | x i − x j | m −| α | for all | α | ≤ m, ≤ i, j ≤ k .Then there exists F ∈ C m ( R n , R D ) with k F k C m ( R n , R D ) ≤ C ( m, n, D ) M and J x ( F ) ∈ H ( x ) for all x ∈ E . We will use the following elementary result regarding semialgebraic functions. For a proof,see [32].
Lemma 3.1
Suppose f : R → R is semialgebraic. Then there exists a polynomial P ( z, x ) on R such that P ( f ( x ) , x ) ≡ . Moreover, for each x ∈ R there exists δ > such that f ( x ) for x ∈ ( x , x + δ ) is given by a convergent Puiseux series. Corollary 3.1
Let F ( x ) be a semialgebraic function of one variable, satisfying | F ( x ) | = O ( x p ) on (0 , c ] for some given p . Then the derivatives of F satisfy | F ( k ) ( x ) | = O ( x p − k ) on (0 , c ′ ] for some c ′ . Similarly, if F ( x ) = o ( x p ) for x in (0 , c ) , then F ( k ) ( x ) = o ( x p − k ) for x in (0 , c ′ ) . More generally, | F ( k ) ( x ) | = O ( | F ( x ) | /x k ) on (0 , c ′ ) . Corollary 3.2
Let F be a semialgebraic function in C mloc (Ω ) , where Ω δ = { ( x, y ) ∈ R :0 ≤ y ≤ x < δ } for δ > . Then for small enough δ , F | Ω δ extends to a C m semialgebraicfunction on R . Sketch of proof.
The result follows in one line from known results, but we sketch anelementary proof.Without loss of generality, we may suppose that J (0 , F = 0. Then ∂ kx F ( x ,
0) = o ( x m − k )for k ≤ m , hence ∂ lx ∂ kx F ( x ,
0) = o ( x m − k − l ) for 0 ≤ k, l ≤ m .We set ˜ F ( x , x ) equal to the m-th degree Taylor polynomial of x F ( x , x ) about x = 0 for each fixed x . The above estimates for derivatives of F show that ˜ F is C m on˜Ω δ = { ( x , x ) : 0 ≤ − x ≤ x ≤ δ } , and its x -derivatives up to order m agree with thoseof F on the x -axis. In particular, J (0 , ˜ F = 0.Similarly, we set F ( x , x ) equal to the m-th degree Taylor polynomial of x F ( x , x )about x = x for each fixed x . Then F is C m on Ω δ = { ( x , x ) : 0 ≤ x ≤ x ≤ x ≤ δ } , and its x -derivatives up to order m agree with those of F on the line x = x . Inparticular, J (0 , F = 0.Setting F + = F on Ω δ ˜ F on ˜Ω δ F on Ω δ , we see that F + is a C m semialgebraic function on { ( x , x ) : x ∈ [0 , δ ] , − x ≤ x ≤ x } , F + = F on Ω δ , and J (0 , F + = 0.Next, let θ ( t ) be a C m semialgebraic function of one variable, equal to 1 in [0 ,
1] andsupported in [ − , δ , the function F ++ ( x , x ) = θ ( x x ) · F + ( x , x )7or x > F ++ ( x , x ) = 0 otherwise, is a C m semialgebraic function on the disc B (0 , δ )that agrees with our given F on Ω δ .Finally, multiplying F ++ by a semialgebraic cutoff function supported in a small discabout (0 ,
0) and equal to 1 in a smaller disc, we obtain a C m semialgebraic function on R that agrees with F on Ω δ for small enough δ . We recall a few standard properties of semialgebraic sets and functions. • Let U ⊂ R n be an open semialgebraic set, and let F : U → R k be semialgebraic. Thenthere exists a semialgebraic subset X ⊂ U of dimension less than n (the “singular set”of F ) such that F is real-analytic on U \ X . (See Chapter 8 in [6].) • A zero-dimensional semialgebraic set is finite. A one-dimensional semialgebraic set isa union of finitely many real-analytic arcs and finitely many points. (See Chapter 2in [6].)
For sets
X, Y , we denote a map Ξ from X to the power set of Y by Ξ : X ⇒ Y and callsuch Ξ a set-valued map; a set-valued map Ξ is semialgebraic if { ( x, y ) : y ∈ Ξ( x ) } is asemialgebraic set. Let E ⊂ R n and Ξ : E ⇒ R D . A selection of Ξ is a map f : E → R D such that f ( x ) ∈ Ξ( x ) for every x ∈ E . We recall the following well-known result regardingsemialgebraic selection (see, for example, [36]). Theorem 3
Let
Ξ : E ⇒ R D be semialgebraic. If each Ξ( x ) is nonempty, then Ξ has asemialgebraic selection. Recall from [30] the following result
Lemma 3.2 (Growth Lemma)
Let E ⊂ R n and E + ⊂ E × R n be compact and semial-gebraic, with dim E + ≥ . Let A be a semialgebraic function on E + . Then there exist aninteger K ≥ , a semialgebraic function A on E , and a compact semialgebraic set E + ⊂ E + ,with the following properties. (GL1) dim E + < dim E + .For x ∈ E , set E + ( x ) = { y ∈ R n : ( x, y ) ∈ E + } and E + ( x ) = (cid:8) y ∈ R n : ( x, y ) ∈ E + (cid:9) .Then, for each x ∈ E , the following hold. (GL2) If E + ( x ) is empty, then | A ( x, y ) | ≤ A ( x ) for all y ∈ E + ( x ) . GL3) If E + ( x ) is non-empty, then | A ( x, y ) | ≤ A ( x ) · (cid:2) dist (cid:0) y, E + ( x ) (cid:1)(cid:3) − K for all y ∈ E + ( x ) \ E + ( x ) . The Growth Lemma follows easily from a special of a theorem of Lojasiewicz and Wachta[35], as explained in [30]. We thank W. Paw lucki for teaching us that implication.We will apply the Growth Lemma to prove the following.
Lemma 3.3
Let F ( x, y ) be a bounded semialgebraic function on [ − , × (0 , , and supposethat (5) lim y → + F ( x, y ) = 0 for each x ∈ [ − , .Then there exist a positive integer N and a semialgebraic function A ( x ) on [ − , such that F ( x, y ) ≤ A ( x ) y N for all ( x, y ) ∈ [ − , × (0 , . Proof.
It is enough to show that for some positive integer N we have(6) sup y ∈ (0 , | F ( x, y ) | y /N < ∞ for all x ∈ [ − ,
1] ,for we may then set A ( x ) = sup y ∈ (0 , | F ( x,y ) | y /N , and A ( x ) will depend semialgebraically on x .For each fixed x , the function y F ( x, y ) is bounded and given near (0 ,
0) by a conver-gent Puiseux series that tends to zero as y → + . Hence, for some positive integer N x wehave(7) sup y ∈ (0 , | F ( x, y ) | y /N x < ∞ .Our task is to show that N x may be taken independent of x. Thanks to (7), we may excludefrom consideration any given finite set of “bad” x ∈ [ − , x, ε ) ∈ [ − , × (0 ,
1] there exists δ ∈ (0 , x, ε, δ ) belongs to the semialgebraic set { ( x, ε, δ ) ∈ [ − , × (0 , × (0 ,
1] : | F ( x, y ) | ≤ ε for all y ∈ (0 , δ ] } . Hence, there exists a semialgebraic function δ ( x, ε ) mapping [ − , × (0 ,
1] into (0 ,
1] suchthat(8) | F ( x, y ) | ≤ ε for y ∈ (0 , δ ( x, ε )] , x ∈ [ − , , ε ∈ (0 , . We set δ ( x,
0) = 1 for x ∈ [ − , δ : [ − , × [0 , → (0 ,
1] is semialgebraic andsatisfies (8).We now apply Lemma 3.2 to the function δ ( x,ε ) .Thus, we obtain a semialgebraic set E ⊂ [ − , × [0 , N, and apositive semialgebraic function δ ( x ) on [ − , dim E ≤ • For x ∈ [ − , E ( x ) = { ε : ( x, ε ) ∈ E } .Then(9) δ ( x, ε ) ≥ δ ( x ) (all ε >
0) if E = ∅ and(10) δ ( x, ε ) ≥ δ ( x ) · [dist ( ε, E ( x ))] N (all ε E ( x )) if E = ∅ .Because dim E ≤ , there are at most finitely many x ∈ [ − ,
1] for which E ( x ) is infinite.As explained above, we may discard those “bad” x , it is enough to prove (6) for all x such that E ( x ) is finite.From now on, we restrict attention to “good” x, i.e., those x for which E ( x ) is finite.Set ε ( x ) = (cid:26) min ( E ( x ) \ { } )1 if E ( x ) contains points other than 0otherwise .So ε ( x ) > x .If E ( x ) = ∅ , then dist ( ε, E ( x )) ≥ ε for 0 < ε ≤ ε ( x ), hence (10) gives(11) δ ( x, ε ) ≥ δ ( x ) ε N for 0 < ε ≤ ε ( x ) .If instead E ( x ) = ∅ , then because ε ( x ) = 1 , (9) again gives (11). Thus, (11) holds in allcases.Now suppose 0 < y < δ ( x ) · ( ε ( x )) N .Then, setting ε = (cid:16) yδ ( x ) (cid:17) /N and applying (11), we find that δ ( x, ε ) ≥ y. The definingproperty of δ ( x, ε ) therefore tells us that | F ( x, y ) | ≤ ε = (cid:18) yδ ( x ) (cid:19) /N .Thus, for any “good” x, we have shown that(12) | F ( x, y ) | y /N ≤ ( δ ( x )) − /N for 0 < y < δ ( x ) · ( ε ( x )) N .On the other hand, recall that F is bounded; say, | F ( x, y ) | ≤ M for all ( x, y ) ∈ [ − , × (0 , | F ( x, y ) | y /N ≤ M ( δ ( x )) /N ε ( x ) for δ ( x ) · ( ε ( x )) N ≤ y ≤ n -dimensional version of Lemma 3.3, but we don’tdiscuss it here. 10 .6 Logarithmic Derivatives of Semialgebraic Functions Let V be a semialgebraic subset of R n × R m . Given x ∈ R n , we write V ( x ) to denote the setof all t ∈ R m such that ( x, t ) ∈ V . Given ( x, t ) ∈ R n × R m , we write δ V ( x, t ) to denote thedistance from t to V ( x ). We take δ V ( x, t ) = + ∞ if V ( x ) is empty. For a smooth function F ( x, t ) on R n × R m , we write ∇ t F ( x, t ) to denote the gradient of the function t F ( x, t ).The following theorem is proven by A. Parusinski in [37, 38]. We thank Edward Bier-stone, Jean-Baptiste Campesato, Pierre Milman, and Wieslaw Paw lucki for pointing out thereferences, and thus helping us remove 10 pages from our paper. Theorem 4
Let F ( x, t ) be a (real-valued) subanalytic function of ( x, t ) ∈ R n × R m . Thenthere exist a closed codimension 1 subanalytic set V ⊂ R n × R m and a constant C > suchthat outside V the function F is smooth and moreover, (14) |∇ t F ( x, t ) | ≤ C | F ( x, t ) | δ V ( x, t ) .If F is semialgebraic, then we can take V to be semialgebraic. As a special case of Theorem 4, we have the following.
Theorem 5
Let F ( x ) be a semialgebraic function on R n . Then there exist a closed semial-gebraic V ⊂ R n of dimension at most ( n − , and a constant C , such that F is C mloc outside V , and |∇ F ( x ) | ≤ C | F ( x ) | · [ dist ( x, V )] − for x ∈ R n \ V . We recall the following result from convex geometry. Surely more precise versions of theresult are well known, but we had trouble tracking down a reference so we will provide aproof.
Theorem 6 (Helly’s Theorem Variant)
Let ( p ω ) ω ∈ Ω be a family of seminorms on a vec-tor space V of dimension D . Assume that sup ω ∈ Ω p ω ( v ) < ∞ for every v ∈ V . Then thereexist ω , · · · , ω L ∈ Ω , with L depending only on D , such that sup ω ∈ Ω p ω ( v ) ≤ C · max { p ω ( v ) , · · · , p ω L ( v ) } for all v ∈ V, with C also depending only on D . We use the following variant of the classical Helly theorem (see Section 3 in [14]) fromelementary convex geometry. 11 emma 3.4
Let ( K ω ) ω ∈ Ω be a collection of compact convex symmetric subsets of R D . Sup-pose the intersection of all the K ω has nonempty interior. Then there exist ω , · · · , ω L suchthat K ω ∩ · · · ∩ K ω L ⊂ C · T ω ∈ Ω K ω , where C and L depend only on D . The proof of the “Lemma on Convex Sets” in Section 3 of [14] applies here and provesLemma 3.4, even though our present hypotheses differ slightly from those of [14].We apply Lemma 3.4 to prove Theorem 6.
Proof of Theorem 6.
Suppose first that each p ω is a norm, not just a seminorm. Thenthe conclusion of Theorem 6 follows by applying Lemma 3.4 to the family of convex sets K ω = { v ∈ V : p ω ( v ) ≤ } , ω ∈ Ω.Now suppose each p ω is a seminorm. Let H ( ω ) = { v ∈ V : p ω ( v ) = 0 } , and let H be theintersection of all the H ( ω ). Each H ( ω ) is a vector subspace of V . Consequently there exist λ , · · · , λ s ∈ Ω, with s ≤ D , such that H = H ( λ ) ∩ · · · ∩ H ( λ s ).For ω ∈ Ω and v ∈ V , set p ∗ ω ( v ) = p λ ( v ) + · · · + p λ s ( v ) + p ω ( v ). Then p ∗ ω is a seminorm on V , and p ∗ ω ( v ) = 0 if and only if v ∈ H . Regarding each p ∗ ω as a norm on V /H , and applyingTheorem 6 for collections of norms, we complete the proof of Theorem 6.
The purpose of this section is to reduce Theorem 1 to the following:
Lemma 4.1 (Main Lemma)
Let H = ( H ( x )) x ∈ R be a semialgebraic bundle for C mloc ( R , R D ) .Assume H is Glaeser stable. Assume H (0) = { } . Then, for small enough c > , H| Γ( c ) hasa semialgebraic section, where Γ( c ) = { ( x , x ) ∈ R : x ∈ [0 , c ] , ≤ x ≤ x } . To deduce Theorem 1 from Lemma 4.1 we argue as follows.Suppose we are given a Glaeser stable bundle H = ( H ( x )) x ∈ R for C mloc ( R , R D ) with H ( x ) ⊂ P D depending semialgebraically on x . Assume H (0) = { } .Let Γ( c ) = { ( x , x ) ∈ R : x ∈ [0 , c ] , ≤ x ≤ x } . Theorem 2 tells us that H| Γ( c ) hasa section F c . The main lemma asserts that for c small enough H| Γ( c ) has a semialgebraicsection.We will cover a full neighborhood of 0 by rotating wedges of the form Γ( c ). Using apartition of unity subordinate to the cover and the fact that H (0) = { } , we can then patchtogether sections of H , and obtain a semialgebraic section over a full neighborhood of 0.We may drop the restriction H (0) = { } , because without loss of generality our givensection F c has jet 0 at the origin, so we may just cut down H (0) to { } . We can also dropthe restriction that H is Glaeser stable (assuming H has a section) since we can always passto the stable Glaeser refinement. Thus, any semialgebraic bundle having a section has asemialgebraic section over some neighborhood of 0. We can use compactness and a partitionof unity to conclude that H admits a semialgebraic section over any given compact set. Lemma 4.2
Suppose H ( z ) depends semialgebraically on z ∈ R . If H = ( H ( z )) z ∈ R hasa section, then H has a section F ∈ C mloc ( R , R D ) such that for all | α | ≤ m , | ∂ α F ( x ) | ≤ C (1 + | x | ) K on R , for some C and K . roof. To prove this lemma, we may assume that H is Glaeser stable.Taking E R = { x ∈ R : | x | ≤ R } with R ≥
1, and applying Theorem 2, we obtain asection F R of H| E R , with || F R || C m ≤ C ( R ) K , because the “ M ” in the result quoted aboveapplied to H| E R can be taken to depend semialgebraically on R . (That’s where we use thefact that the bundle H is semialgebraic.)We can now easily use a partition of unity to patch together F k , k = 1 , , , · · · , into asection F as in the conclusion of Lemma 4.2.Fix K as in the conclusion of Lemma 4.2. Let Φ : Open Disc ∆ → R be a semialgebraicdiffeomorphism, for example, Φ( x ) = x −| x | . Let θ ( x ) > R that tends to zero so rapidly that ∂ α [( θF ) ◦ Φ]( y ) →
0, for all | α | ≤ m as y → ∂ ∆ , whenever | ∂ α F ( x ) | ≤ C (1 + | x | ) K on R , | α | ≤ m .We can now form a bundle H ∗ as follows: For x in ∆, the fiber H ∗ ( x ) consists of all J x (( θF ) ◦ Φ) for sections F of the bundle H .The fibers of H ∗ over points not in ∆ are { } .Then H ∗ is a semialgebraic bundle admitting a section.We have seen that semialgebraic bundles with sections have semialgebraic sections overany compact set. In particular, H ∗ has a semialgebraic section F over ∆ closure . Then F◦ Φ − ( x ) θ ( x ) is a semialgebraic section of H over R .Consequently, we can deduce Theorem 1 from Lemma 4.1.The rest of the paper is devoted to the proof of Lemma 4.1. Fix U ⊂ R n open, semialgebraic. Fix ψ : U → R k Nash. Let ˆ ψ ( x ) = ( x, ψ ( x )) ∈ R n × R k for x ∈ U . We set ˆ U = ˆ ψ ( U ). Let P denote the vector space of polynomials of degree at most m on R n × R k . We write z = ( x, y ) to denote a point of R n × R k . We write R z to denotethe ring obtained from P by multiplication of m -jets at z . We fix a bundle H = ( H ( z )) z ∈ ˆ U ,where, for each z = ˆ ψ ( x ) ∈ ˆ U we have H ( z ) = f x + I ( x ), f x ∈ P D , I ( x ) an R ˆ ψ ( x ) -submoduleof P D . (We point out that H is a bundle, not a classical bundle, see Remark 1.1.)We suppose H is Glaeser stable. We assume that H ( z ) depends semialgebraically on z ∈ ˆ U . (We sometimes abuse notion by writing I ( z ) for I ( x ), where z = ˆ ψ ( x ).)Under the above assumptions and definitions, we will prove the following result. Lemma 5.1
There exist a semialgebraic set U bad ⊂ R n of dimension less than n ; Nashfunctions A ijβ , G i on U \ U bad ( i = 1 , · · · , i max , j = 1 , · · · , D, β a multiindex of order ≤ m for R k ) with the following property. Let B ⊂ U \ U bad be a closed ball. Set ˆ B = ˆ ψ ( B ) .Let F = ( F , · · · , F D ) ∈ C mloc ( R n × R k , R D ) . Then F is a section of H| ˆ B if and only if P | β |≤ m P Dj =1 A ijβ ( x ) · ( ∂ βy F j ( x, ψ ( x ))) = G i ( x ) for all x ∈ B (each i ). roof. We may suppose that f x and I ( x ) depend semialgebraically on x ∈ U . We write f x = ( f x , · · · , f xD ) and ψ ( x ) = ( ψ ( x ) , · · · , ψ k ( x )) ( x ∈ U ).For l = 1 , · · · , n , we introduce the vector field X l = ∂∂x l + k X p =1 ∂ψ p ( x ) ∂x l ∂∂y p on U × R k . On U × R k , then X l are Nash, and [ X l , X l ′ ] = 0. For α = ( α , · · · , α n ), we write X α = X α · · · X α n n .The X , · · · , X n , ∂∂y , · · · , ∂∂y k form a frame on U × R k . Because I ( x ) depends semialge-braically on x ∈ U , we may express(15) I ( x ) = ( ( P , · · · , P D ) ∈ P D : P | α | + | β |≤ mj =1 , ··· ,D ˜ A ijαβ ( x ) (cid:0) X α ∂ βy P j (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ˜ ψ ( x ) = 0, for i = 1 , · · · , i max ) for semialgebraic ˜ A ijαβ on U .We take U to be the union of the singular sets of the ˜ A ijαβ . Then U is a semialgebraicset of dimension < n in R n , and the ˜ A ijαβ are real-analytic on U \ U .We may therefore rewrite the equation in (15) in the form X | α | + | β |≤ mj =1 , ··· ,D (cid:0) X α (cid:8) A ijαβ ( x ) ∂ βy P j (cid:9)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ ψ ( x ) = 0.The A ijαβ are Nash on U \ U . Thus, for any closed ball B ⊂ U \ U the following holds.(We set ˆ B = ˆ ψ ( B ).)A given F = ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) is a section of ( I ( z )) z ∈ ˆ B if and only if X | α |≤ m X α X | β |≤ m −| α | A ijαβ ( x ) ∂ βy F j ( x, y ) = 0 on ˆ B for all i .We look for integers s ≥ A ijαβ on U \ U withthe following property (“Property Q ( s )”):Let B ⊂ U \ U be a closed ball; set ˆ B = ˆ ψ ( B ). Then ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) is a section of ( I ( z )) z ∈ ˆ B if and only if(16) X | α |≤ s X α X | β |≤ m −| α | D X j =1 A ijαβ ( x ) ∂ βy F j ( x, y ) = 0 on ˆ B for all i .We have seen that we can achieve Property Q ( m ).14 laim 5.1 Let s be the smallest possible integer ≥ for which we can achieve Property Q ( s ) , and let A ijαβ be as in Property Q ( s ) . Then s = 0 . In other words, Property Q (0) holds. Proof of Claim 5.1.
Assuming s ≥
1, we will achieve Property Q ( s − s is as small as possible.Fix B ⊂ U \ U a closed ball, and let ( F , · · · , F D ) ∈ C mloc ( R n × R k , R D ) be a section of( I ( z )) z ∈ ˆ B . (As always, ˆ B = ψ ( B ).) Fix x ∈ B and fix a multiindex α with | α | = s . For j = 1 , · · · , D , define functions on R n × R k by setting F j ( z ) = θ · F j ( z ) where θ ∈ C ∞ ( R n × R k )with jet ( J ˆ ψ ( x ) θ )( x, y ) = ( x − x ) α .Then ( F , · · · , F D ) ∈ C mloc ( R n × R k , R D ) is a section of ( I ( z )) z ∈ ˆ B because each I ( z ) is an R z -submodule of R Dz .Applying Property Q ( s ) to ( F , · · · , F D ), we learn that X | β |≤ m −| α | D X j =1 A ijα β ( x ) (cid:0) ∂ βy F j (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ ψ ( x ) = 0 (all i ) .This holds for all x and for all | α | = s . Thus, if ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) is asection of ( I ( z )) z ∈ ˆ B , then(17) X | β |≤ m −| α | D X j =1 A ijαβ ( x ) ∂ βy F j ( x, y ) = 0on ˆ B for all | α | = s and for all i . Because the X j are tangent to ˆ B , it follows from (17) that(18) X α X | β |≤ m −| α | D X j =1 A ijαβ ( x ) ∂ βy F j ( x, y ) = 0on ˆ B for all | α | = s and for all i . From (16) and (18), we conclude that(19) X | α |≤ s − X α X | β |≤ m −| α | D X j =1 A ijαβ ( x ) ∂ βy F j ( x, y ) = 0on ˆ B for all i . Thus, any section of ( I ( z )) z ∈ ˆ B satisfies (17) and (19). Conversely, suppose( F , · · · , F D ) ∈ C mloc (cid:0) R k × R k , R D (cid:1) satisfies (17) and (19). Then, because (17) implies (18),it follows that (16) holds, and consequently ( F , · · · , F D ) is a section of ( I ( z )) z ∈ ˆ B . Thus,a given ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) is a section of ( I ( z )) z ∈ ˆ B if and only if (17) and(19) hold. If s ≥ , this implies that we have achieved Property Q ( s − s , and establishing Claim 5.1.We return to the proof of Lemma 5.1. Because Property Q ( s ) holds with s = 0, thereexist Nash functions A ijβ on U \ U , for which the following (“Property Q ∗ ”) holds:15et B ⊂ U \ U be a closed ball. Set ˆ B = ˆ ψ ( B ). Then a given ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) is a section of ( I ( z )) z ∈ ˆ B if and only if(20) X | β |≤ m D X j =1 A ijβ ( x ) ∂ βy F j ( x, y ) = 0 on ˆ B (all i ) . We fix A ijβ as above.We now return to our bundle H = ( f z + I ( z )) z ∈ ˆ U .(We abuse notation by writing f z for f x where z = ˆ ψ ( x ).)Let B ⊂ U \ U be a closed ball, and let ˆ B = ˆ ψ ( B ). Let ( F , · · · , F D ) and (cid:16) ˜ F , · · · , ˜ F D (cid:17) ∈ C mloc (cid:0) R n × R k , R D (cid:1) be any two sections of H| ˆ B .Then (cid:16) F − ˜ F , · · · , F D − ˜ F D (cid:17) is a section of ( I ( z )) z ∈ ˆ B , and therefore by (20), we have(21) X | β |≤ mj =1 , ··· ,D A ijβ ( x ) ∂ βy F j ( x, y ) = X | β |≤ mj =1 , ··· ,D A ijβ ( x ) ∂ βy ˜ F j ( x, y ) on ˆ B for all i .Moreover, given x ∈ B , we can take our section (cid:16) ˜ F , · · · , ˜ F D (cid:17) above to satisfy J ˆ ψ ( x ) ˜ F j = f x j ( j = 1 , · · · , D ) ,because ( f x , · · · , f x D ) ∈ H (cid:16) ˆ ψ ( x ) (cid:17) and H| ˆ B is Glaeser stable and has nonempty fibers.(See Theorem 2.) Therefore, (21) implies that(22) X | β |≤ m D X j =1 A ijβ ( x ) ∂ βy F j ( x, y ) = G i ( x )on ˆ B for each i , where G i ( x ) = X | β |≤ m D X j =1 A ijβ ( x ) (cid:0) ∂ βy f x (cid:1) | ˆ ψ ( x ) (cid:0) x ∈ U \ U (cid:1) .Clearly, G i ( x ) is a semialgebraic function on U \ U , and it is independent of the ball B in the above discussion.Thus, we have seen that any section ( F , · · · , F D ) of H| ˆ B must satisfy (22).Conversely, suppose ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) satisfies (22). Let (cid:16) ˜ F , · · · , ˜ F D (cid:17) ∈ C mloc (cid:0) R n × R k , R D (cid:1) be a section of H| ˆ B . (We know that a section exists because H| ˆ B isGlaeser stable and has nonempty fibers.) We know that (cid:16) ˜ F , · · · , ˜ F D (cid:17) satisfies (22), hence X | β |≤ m D X j =1 A ijβ ( x ) ∂ βy h F j − ˜ F j i ( x, y ) = 016n ˆ B for each i .Recalling Property Q ∗ , we now see that (cid:16) F − ˜ F , · · · , F D − ˜ F D (cid:17) is a section of ( I ( z )) z ∈ ˆ B . Because (cid:16) ˜ F , · · · , ˜ F D (cid:17) ∈ C mloc (cid:0) R n × R k , R D (cid:1) is a section of H| ˆ B = ( f z + I ( z )) z ∈ ˆ B , we con-clude that ( F , · · · , F D ) is a section of H| ˆ B . Thus, if ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) satisfies (22), then it is a section of H| ˆ B .We have now seen that a given ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) is a section of H| ˆ B ifand only if (22) holds.Thus, all the conclusions of Lemma 5.1 hold, except that perhaps the G i are not real-analytic.We set U =union of all the singular sets of the semialgebraic functions G i . That’s asemialgebraic set of dimension < n in R n .We take U bad = U ∪ U , a semialgebraic set of dimension < n in R n .The functions A ijβ and G i are Nash on U \ U bad .If B ⊂ U \ U bad is a closed ball and ˆ B = ψ ( B ), then a given ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) is a section of H| ˆ B if and only if X | β |≤ m D X j =1 A ijβ ( x ) (cid:0) ∂ βy F j (cid:1) | ˆ ψ ( x ) = G i ( x )on B for each i .This completes the proof of Lemma 5.1 . Remark 5.1
Lemma 5.1 and its proof hold also for k = 0 . In that case, ˆ ψ is the identitymap and there are no y -variables, hence no y -derivatives in the conclusion of Lemma 5.1. Corollary 5.1
Let H , U, ψ, · · · be as in Lemma 5.1. Let ( F , · · · , F D ) ∈ C mloc (cid:0) R n × R k , R D (cid:1) . Then ( F , · · · , F D ) is a section of H| ˆ U \ ˆ ψ ( U bad ) if and only if X | β |≤ m D X j =1 A ijβ ( x ) ∂ βy F j ( x, y ) = G i ( x ) on ˆ U \ ˆ ψ ( U bad ) , for all i . Proof. U \ U bad is a union of (infinitely many overlapping) closed balls B . Applying Lemma5.1 to each B , we obtain the desired conclusion. Suppose we are given a system of linear equations(23) X i + P j>k A ij X j = b i , for i = 1 , · · · , k with | A ij | ≤ k for i = 1 , · · · k, j = k + 1 , · · · , M ,and 1724) P j>k C ij X j = g i , for i = k + 1 , · · · , N ,where 0 ≤ k ≤ N, M ; the A ij , C ij , b i , g i are semialgebraic functions defined on a semialge-braic set E ⊂ R n ; and X , · · · , X M are unknowns.We say that this system is in k -echelon form on E If k = 0, then we have simply (24) for i = 1 , · · · , N , so every system of linear equationswith coefficient matrix and right-hand sides depending semialgebraically on x ∈ E is in0-echelon form on E .If also C ij ≡ E for all i = k + 1 , · · · , N , j = k + 1 , · · · , M , then we say that oursystem of equations is in echelon form on E . In particular, a system in k -echelon form with k = min { N, M } is in echelon form on E . Suppose our system is in k -echelon form with k < min { N, M } . We partition E as follows. Let E good = { x ∈ E : All the C ij ( x ) = 0 } . For˜ i = k + 1 , · · · , N and ˜ j = k + 1 , · · · , M , we let ˜ E (˜ i, ˜ j ) = { x ∈ E : | C ˜ i ˜ j | = max ij | C ij | > } .The E good and ˜ E ( i, j ) form a covering of E .We enumerate the pairs ( i, j ) in any order and then form sets E ( i, j ) by removing from˜ E ( i, j ) all points contained in some ˜ E ( i ′ , j ′ ) with ( i ′ , j ′ ) preceding ( i, j ). Then E good andthe E ( i, j ) form a partition of E into semialgebraic sets. On E good , our system is in echelonform.On each E ( a, b ), we will exhibit a system of linear equations in ( k + 1)-echelon form,equivalent to the given system (23), (24). For fixed ( a, b ), we relabel equations and unknownsso that our system still has the form (23), (24), but with | C k +1 ,k +1 | = max ij | C ij | > C k +1 ,k +1 , we may assume that(25) C k +1 ,k +1 = 1and all(26) | C ij | ≤ . Note that A ij , C ij , b i , g i still depend semialgebraically on x . From each equation (23), wesubtract A i ( k +1) times equation (24) with i = k + 1. From each equation (24) ( i = k + 1), wesubtract C i,k +1 times equation (24) with i = k + 1. Thus, we obtain equations of the form(27) X i + P j>k ˜ A ij X j = ˜ b i , for i = 1 , · · · , kX k +1 + P j>k +1 C k +1 ,j X j = g k +1 , P j ≥ k +1 ˜ C ij X j = ˜ g i , for i > k + 1 . Here, ˜ A ij = A ij − A i ( k +1) C k +1 ,j for i = 1 , · · · , k , j ≥ k + 1; and ˜ C ij = C ij − C i,k +1 C k +1 ,j for i = k + 2 , · · · , N , j > k + 1.In particular, ˜ A i,k +1 = A i,k +1 − A i,k +1 · C k +1 ,k +1 = 0, and ˜ C i,k +1 = C i,k +1 − C i,k +1 · C k +1 ,k +1 = 0, thanks to (25).Also, (cid:12)(cid:12)(cid:12) ˜ A ij (cid:12)(cid:12)(cid:12) ≤ | A ij | + | A i,k +1 | · | C k +1 ,j | ≤ | A ij | + | A i,k +1 | (by (26)) ≤ k + 2 k (because oursystem (23), (24) is in k -echelon form)= 2 k +1 . Recall that | C k +1 ,j | ≤ k + 1)-echelon form.We repeat this procedure, starting with a system in 0-echelon form, and partition E more and more finely into pieces E ν , on each of which an equivalent system to (23), (24) iseither in echelon form, or in k -echelon form for ever higher k . The procedure has to stopafter at most min ( N, M ) steps, because a system in k -echelon form with k = min ( N, M ) isautomatically in echelon form.Thus, we have proven the following result
Lemma 5.2
Consider a system of linear equations (28) M X j =1 C ij ( x ) X j = g i ( x ) ( i = 1 , · · · , N ) where the C ij ( x ) and g i ( x ) are semialgebraic functions defined on a semialgebraic set E ⊂ R n .Then we can partition E into semialgebraic sets E ν ( ν = 1 , · · · , ν max ) , for which thefollowing holds for each ν :There exist a permutation π : { , · · · , M } → { , · · · , M } and an integer ≤ k ≤ min ( N, M ) such that for each x ∈ E ν , the system (28) is equivalent to a system of the form (29) (cid:20) X πi + P j>k ˜ A ij ( x ) X πj = ˜ g i ( x ) for i = 1 , · · · , k b i ( x ) for i = k + 1 , · · · , N .That is, for each x ∈ E ν and each ( X , · · · , X M ) ∈ C M , (28) holds at x if and only if (29) holds at x . Here, the ˜ A ij , ˜ g i , and ˜ b i are semialgebraic functions on E ν , and (cid:12)(cid:12)(cid:12) ˜ A ij ( x ) (cid:12)(cid:12)(cid:12) ≤ k on E ν . In essence, the method for solving the system (28) is just the usual Gaussian elimination,except that we take extra care to maintain the growth condition (cid:12)(cid:12)(cid:12) ˜ A ij ( x ) (cid:12)(cid:12)(cid:12) ≤ k . We work with a semialgebraic bundle H = ( H ( x )) x ∈ R . Each H ( x ) is a coset of an R x -submodule of ( R x ) D , depending semialgebraically on x . Here, R x is the ring of the m -jets offunctions at x . A function F = ( F , · · · , F D ) ∈ C mloc (Ω , R D ) (Ω ⊂ R open) is a section of H if for all x ∈ Ω the m -jet J x F belongs to H ( x ). A function F ∈ C mloc (Ω , R D ) is called a localsection near x ( x ∈ Ω) if for some small disc B ⊂ Ω centered at x we have J x F ∈ H ( x )for all x ∈ B .Let Ω = { ( x, y ) ∈ R : 0 ≤ y ≤ x } . Let H = ( H ( x )) x ∈ R be a semialgebraic bundle,with H ((0 , { } . We assume that H has a section. We want a convenient condition onfunctions F ∈ C mloc (Ω , R D ) that is equivalent to the assertion that F | B ∩ Ω interior is a section of H for a small enough disc B centered at the origin. We achieve (approximately) that.19o do so, we partition Ω into semialgebraic open subsets of R , finitely many semialgebraiccurves in R , and finitely many points. To start with, we partition Ω into the point (0 , { ( x,
0) : x > } , { ( x, x ) : x > } , and Ω interior .As we proceed, we will cut up each of our semialgebraic open sets into finitely manysemialgebraic open subsets, finitely many semialgebraic arcs, and finitely many points. Wewon’t keep track explicitly of the arcs and points at first; we just discard semialgebraicsubsets of R of dimension ≤ k = 0 to Ω interior and H . (See Remark 5.1.)Thus, we obtain a semialgebraic V ⊂ Ω interior of dimension ≤
1, outside of which thefollowing holds for some semialgebraic functions A ij ( x ) , φ i ( x ) for 1 ≤ i ≤ i max , ≤ j ≤ D, x ∈ Ω interior \ V :Let F = ( F , · · · , F D ) belong to C mloc ( U, R D ) where U is a neighborhood of x ∈ Ω interior \ V . Then F is a local section of H near x if and only if(30) P Dj =1 A ij ( x ) F j ( x ) = φ i ( x ), for i = 1 , · · · , i max , for all x in a neighborhood of x .The equations (30) have a solution for each fixed x , because H has a section. Next, weapply Lemma 5.2 to the above system of linear equations.Thus, we obtain a partition of Ω interior \ V into semialgebraic sets E ν ( ν = 1 , · · · , ν ),for which we have integers ˜ k ν ≥
0, permutations ˜ π ν : { , · · · , D } → { , · · · , D } , and semi-algebraic functions ˜ A νij ( x ) (1 ≤ i ≤ ˜ k ν , ˜ k ν + 1 ≤ j ≤ D, x ∈ E ν ), ˜ φ νi ( x ) such that for any x ∈ E ν , the system of equations (30) is equivalent to(31) F π ν i ( x ) + X j> ˜ k ν ˜ A νij ( x ) F π ν j ( x ) = ˜ ϕ νi ( x ) for i = 1 , · · · , ˜ k ν . Moreover, the ˜ A νij ( x ) are bounded. Note that the functions ˜ b i in (29) are identically 0because our equations (30) have a solution.Because H has a section, there exists F = ( F , · · · , F D ) ∈ C mloc (cid:0) Ω , R D (cid:1) satisfying (30)for all x ∈ Ω interior \ V , hence also satisfying (31) in E ν . Consequently, the left-hand side of(31) is bounded (for bounded x ), and thus also the ˜ ϕ Di ( x ) are bounded (for bounded x ).Applying Theorem 5, we obtain a semialgebraic V ⊂ R of dimension ≤
1, satisfying(32) | ∂ α ˜ ϕ νi ( x ) | , (cid:12)(cid:12)(cid:12) ∂ α ˜ A νij ( x ) (cid:12)(cid:12)(cid:12) ≤ C [dist ( x, V )] −| α | for bounded x outside V , for | α | ≤ m + 100.By adding ∂ Ω to V and removing from V all points outside Ω, we may assume V ⊂ Ω.(This operation does not increase the distance from V to any point of Ω.)Let ˆ E ν ( ν = 1 , · · · , ν max ) be the connected components of the interiors of the sets E ν \ V ( ν = 1 , · · · , ν ).Then Ω is partitioned into the ˆ E ν and V , where V is a semialgebraic subset of Ω ofdimension ≤
1. The ˆ E ν are pairwise disjoint open connected semialgebraic sets. Any pathin Ω that does not meet V stays entirely in a single ˆ E ν . Indeed, suppose not: let γ ( t ) ∈ Ω( t ∈ [0 , γ (0) ∈ ˆ E ν not staying in ˆ E ν and not meeting V . Pick t ∗ =20nf n t > γ ( t ) ˆ E ν o . Then t ∗ > E ν is open. We can’t have γ ( t ∗ ) ∈ ˆ E ν ′ with ν ′ = ν else γ ( t ) ∈ ˆ E ν ′ (and ∈ ˆ E ν ) for t ∈ [ t ∗ − ε, t ∗ ). We can’t have γ ( t ∗ ) in E ν , since thatwould imply γ ( t ) in E ν for all t in [ t ∗ , t ∗ + ε ]. Thus, γ ( t ∗ ) ∈ V , contradicting the fact that γ does not meet V .Moreover, there exist integers ˆ k ν ≥
0, permutations ˆ π ν : { , · · · , D } → { , · · · , D } ,and semialgebraic functions ˆ A νij ( x ) (cid:16) ≤ i ≤ ˆ k ν , ˆ k ν + 1 ≤ j ≤ D (cid:17) and ˆ ϕ νi ( x ) (cid:16) ≤ i ≤ ˆ k ν (cid:17) defined on ˆ E ν , with the following properties(33) (cid:12)(cid:12)(cid:12) ∂ α ˆ A νij ( x ) (cid:12)(cid:12)(cid:12) , | ∂ α ˆ ϕ νi ( x ) | ≤ C [dist ( x, V )] −| α | for bounded x ∈ ˆ E ν , | α | ≤ m + 100, and(34) Let x ∈ ˆ E ν and let F = ( F , · · · , F D ) be C mloc in a neighborhood of x . Then F is alocal section of H near x if and only if F π ν i ( x ) + X j> ˆ k ν ˆ A νij ( x ) F π ν j ( x ) = ˆ ϕ νi ( x )in a neighborhood of x for each i = 1 , · · · , ˆ k ν .We partition V ∪ { ( x,
0) : x ≥ } ∪ { ( x, x ) : x ≥ } into finitely many Nash open arcs(not containing their endpoints) and finitely many points.For small enough δ > B (0 , δ ) ⊂ R avoids all the above arcs not containing 0 in theirclosure, and all the above points except possibly for the point 0. Taking δ small, we mayassume that the remaining arcs have convergent Puiseux series in B (0 , δ ).Notice that our semialgebraic one-dimensional sets are all contained in Ω; so no arcshave tangent lines at 0 lying outside the sector Ω. Thus, the remaining arcs have the form { y = ψ s ( x ) } in B (0 , δ ), where ψ , · · · , ψ s max are semialgebraic functions of one variable, withconvergent Puiseux expansion in [0 , δ ]. We discard duplicates, i.e., we may assume ψ s isnever identically equal to ψ s ′ for s ′ = s . Note that the line segments { ( x,
0) : 0 < x < δ } and { ( x, x ) : 0 < x < δ } are among our arcs γ s . Taking δ > s = s ′ , either ψ s ( x ) < ψ s ′ ( x ) for all x ∈ (0 , δ ), or ψ s ( x ) > ψ s ′ ( x ) for all x ∈ (0 , δ ).(That’s because the ψ s are given by convergent Puiseux expansions.) Thus, in B (0 , δ ), ourcurves may be labelled so that 0 ≡ ψ ( x ) < ψ ( x ) < · · · < ψ s max ( x ) ≡ x for x ∈ (0 , δ ). Thearcs are γ s = { ( x, ψ s ( x )) : x ∈ [0 , δ ] } for s = 0 , · · · , s max . (Here we have thrown in the point0, and taken δ small to allow ourselves to include x = δ , not just x < δ .)The sets we discarded in passing from V to the semialgebraic arcs γ , · · · , γ s max areirrelevant in the sense that V ∩ B (0 , δ ) ⊂ ( γ ∪ γ ∪ · · · ∪ γ s max ) ∩ B (0 , δ ).Let E s ( s = 1 , · · · , s max ) be the part of the B (0 , δ ) lying between γ s − and γ s , i.e., E s = { ( x, y ) ∈ B (0 , δ ) : 0 < x < δ, ψ s − ( x ) < y < ψ s ( x ) } .Any two points in a given E s may be joined by a path in B (0 , δ ) \ S s max s =0 γ s ⊂ B (0 , δ ) \ V ,hence all points in a given E s lie in the same ˆ E ν .Therefore, for s = 1 , · · · , s max , there exist k s ≥
0, permutations π s : { , · · · , D } →{ , · · · , D } , and semialgebraic functions A sij ( x ), ψ si ( x ) (1 ≤ i ≤ k s ; j = k s + 1 , · · · , D ) on E s ,with the following properties 2135) Let x ∈ E s , and let F = ( F , · · · , F D ) be C mloc in a neighborhood of x . Then F is alocal section of H near x if and only if(36) F π s i ( x ) + P j>k s A sij ( x ) F π s j ( x ) = ψ si ( x ) in a neighborhood of x for each i = 1 , · · · , k s .Moreover,(37) | ∂ α A sij ( x ) | , | ∂ α ψ si ( x ) | ≤ C [dist( x, γ s ∪ γ s − )] −| α | on E s for | α | ≤ m + 100.In particular, if F = ( F , · · · , F D ) ∈ C mloc (Ω , R D ), then J x F ∈ H ( x ) for all x ∈ [Ω ∩ B (0 , δ )] \ ( γ ∪ · · · ∪ γ s max ) if and only if for each s = 1 , · · · , s max , (36) holds on all of E s .Next, we apply Lemma 5.1 to H s = ( H ( x )) x ∈ γ s , ( s = 0 , · · · , s max ). We obtain semialge-braic functions for which the following holds.Let ( x , ψ s ( x )) ∈ γ s be given, and let F = ( F , · · · , F D ) ∈ C mloc (cid:0) U, R D (cid:1) , where U isa neighborhood of γ s in R . Then, except for finitely many bad x , we have the followingequivalence: F is a local section of H s near ( x , ψ s ( x )) if and only if X ≤ j ≤ D ≤ l ≤ m Θ isjl ( x ) ∂ ly F j | ( x,ψ s ( x )) = g si ( x ) ( i = 1 , · · · , i max ( s ))for all x in a neighborhood of x . Here, the Θ’s and g ’s are semialgebraic functions of onevariable. To say that F is a local section of H s near ( x , ψ s ( x )) means that J ( x,ψ s ( x )) F ∈ H ( x, ψ s ( x )) for all x in a neighborhood of x .By restricting attention to B (0 , δ ) and taking δ > B (0 , δ )all these bad x , except for x = 0.Combining our results (35), (37) on the E ν with the above result on the arcs γ s , we obtainthe following result. Lemma 5.3
Let
Ω = { ( x, y ) ∈ R : 0 ≤ y ≤ x ≤ } and let H = ( H ( x )) x ∈ Ω be a semialge-braic bundle, with each H ( x ) consisting of m -jets at x of functions from R to R D .Assume H ((0 , { } and assume H has a section.Then there exist the following objects, with properties to be specified below: • A positive number δ ∈ (0 , . • Semialgebraic functions ψ ( x ) < ψ ( x ) < · · · < ψ s max ( x ) = x on (0 , δ ) , all givenby convergent Puiseux expansions on (0 , δ ) . • Integers k s (0 ≤ k s ≤ D ) and permutations π s : { , · · · , D } → { , · · · , D } for s =1 , · · · , D . • Semialgebraic functions A sij ( x, y ) ( s = 1 , · · · , s max , ≤ i ≤ k s , k s < j ≤ D ) and ϕ si ( x, y )( s = 1 , · · · , s max , ≤ i ≤ k s ) defined on E s = { ( x, y ) : 0 < x < δ, ψ s − ( x ) < y < ψ s ( x ) } . Semialgebraic functions Θ sijl ( x ) , g si ( x ) ( s = 0 , · · · , s max , i = 1 , · · · , i max ( s ) , j = 1 , · · · , D,l = 0 , · · · , m defined on (0 , δ ) , and given there by there by convergent Puiseux expan-sions.The above objects have the following properties • (Estimates) For ( x, y ) ∈ Ω with < x < δ and ψ s − ( x ) < y < ψ s ( x ) , we have (cid:12)(cid:12) ∂ α A sij ( x, y ) (cid:12)(cid:12) , | ∂ α ϕ si ( x, y ) | ≤ C [min ( | y − ψ s ( x ) | , | y − ψ s − ( x ) | )] −| α | for | α | ≤ m + 100 . • (Condition for sections) Let F = ( F , ..., F D ) ∈ C mloc (Ω , R D ) , and suppose J x F ∈ H ( x ) for all x ∈ Ω .(38) Then for s = 1 , · · · , s max , i = 1 , · · · , k s , x ∈ (0 , δ ) , ψ s − ( x ) < y < ψ s ( x ) , we have F π s i ( x, y ) + X D ≥ j>k s A sij ( x, y ) F π s j ( x, y ) = ϕ si ( x, y ) ;and for s = 0 , , · · · , s max , i = 1 , · · · , i max ( s ) , x ∈ (0 , δ ) , we have D X j =1 m X l =0 Θ sijl ( x ) ∂ ly F j ( x, ψ s ( x )) = g si ( x ) ;and J (0 , F j = 0 for all j .Conversely, if F = ( F , ..., F D ) ∈ C mloc (Ω , R D ) and the conditions in (38) are satisfied,then J z F ∈ H ( z ) for all z = ( x, y ) ∈ Ω with ≤ x < δ . This section is devoted to the proof of the following lemma. See (A) and (B) in the Intro-duction.
Lemma 6.1 (Second Main Lemma)
Let H = ( H ( z )) z ∈ Ω with Ω = { ( x, y ) ∈ R : 0 ≤ y ≤ x ≤ } and suppose H ( z ) depends semialgebraically on z . (As usual, H ( z ) ⊂ R Dz is a coset of an R z -submodule.)Suppose H has a section, and suppose H ((0 , { } . Then there exist semialgebraicfunctions θ sijl ( x ) , g si ( x ) , ˜ θ sijl ( x ) , ˜ g si ( x ) of one variable, and ψ ( x ) < · · · < ψ s max ( x ) = x ,also semialgebraic, for which the following hold.Suppose F = ( F , · · · , F D ) ∈ C m (Ω , R D ) is a section of H . Let f sjl ( x ) = ∂ ly F j ( x, ψ s ( x )) for ≤ s ≤ s max , ≤ l ≤ m , ≤ j ≤ D .Then(39) P j,l θ sijl ( x ) f sjl ( x ) = g si ( x ) on (0 , δ ) for some δ > for each s, i ; and P j,l ˜ θ sijl ( x ) f sjl ( x ) =˜ g si ( x ) + o (1) as x → + , each s , i ; and f sjl ( x ) = P m − lk =0 1 k ! f s − j ( l + k ) ( x ) · ( ψ s ( x ) − ψ s − ( x )) k + o (cid:16) [ ψ s ( x ) − ψ s − ( x )] m − l (cid:17) as x → + , each s , j , l .
40) Conversely, if f sjl ( x ) are semialgebraic functions satisfying (39), then there exists asemialgebraic C m section F = ( F , · · · , F D ) of H over Ω δ ′ = { ( x, y ) : 0 ≤ y ≤ x ≤ δ ′ } (some δ ′ > ) such that ∂ ly F j ( x, ψ s ( x )) = f sjl ( x ) for < x < δ ′ . We call the curves y = ψ s ( x ) “critical curves”. Fix m ≥
1. Recall that P denotes the space of polynomials of degree ≤ m on R , and J z F ∈ P denotes the m -jet of F at z ∈ R . ⊙ z denotes multiplication of jets at z . We write p to denote the space of polynomials of degree ≤ m on R . If F ( x, y ) is a C mloc function in aneighborhood of (¯ x, j ¯ x F ∈ p is the m -jet at 0 of the function y F (¯ x, y ). We write ⊡ to denote multiplication of m -jets at 0 of C mloc functions of one variable.If ~F = ( F , · · · , F j max ) is a vector of C mloc functions on R , then J z ~F denotes( J z F , · · · , J z F j max ) ∈ P j max . Similarly, j ¯ x ~F denotes ( j ¯ x F , · · · , j ¯ x F j max ) ∈ p j max . A function F : (0 , δ ) → p may be regarded as a function of ( x, y ) ∈ (0 , δ ) × R such thatfor fixed x , the function y F ( x, y ) is a polynomial of degree at most m .Fix positive integers i max , j max . Let Aff denote the vector space of all affine functionsdefined on p j max + i max . We make the following assumptions: • We are given C ∞ semialgebraic functions A ij , B i , ( i = 1 , · · · , i max , j = 1 , · · · , j max )defined on Ω , where for δ >
0, Ω δ = { ( x, y ) ∈ R : 0 < x < δ, < y < ψ ( x ) } , and ψ : (0 , → (0 , ∞ ) is a semialgebraic function satisfying 0 < ψ ( x ) ≤ x for x ∈ (0 , • We assume that ∂ α A ij , ∂ α B i extend to continuous functions on Ω +1 for | α | ≤ m , where,for δ >
0, Ω + δ = { ( x, y ) ∈ R : 0 < x ≤ δ, < y ≤ ψ ( x ) } . • We suppose that | ∂ α A ij ( x, y ) | ≤ Cy −| α | , and | ∂ α B i ( x, y ) | ≤ Cy −| α | on Ω +1 for | α | ≤ m . Lemma 6.2
Under the above assumptions, there exist δ ∈ (0 , and semialgebraic maps λ , · · · , λ k max , µ , · · · , µ l max : (0 , δ ) → Aff such that the following hold:(41) Suppose ~F = ( F , · · · , F j max ) and ~G = ( G , · · · , G i max ) belong to C m (Ω closure δ , R j max ) and C m (Ω closure δ , R i max ) respectively, with J (0 , ~F = 0 , J (0 , ~G = 0 . Suppose also that G i = P j A ij F j + B i for each i . Then [ λ k (¯ x )]( j ¯ x ~F , j ¯ x ~G ) = 0 for k = 1 , · · · , k max , ¯ x ∈ (0 , δ ) , and [ µ l (¯ x )]( j ¯ x ~F , j ¯ x ~G ) is bounded on (0 , δ ) and tends to zero as ¯ x → , for each l = 1 , · · · , l max . We do not assume ~F or ~G is semialgebraic.
42) Suppose there exists an ( ~F , ~G ) as in (41) . Let ~F = ( F , · · · , F j max ) , ~G = ( G , · · · , G i max ) ,where the F j and G i are semialgebraic maps from (0 , δ ) → p . Suppose that [ λ k (¯ x )]( ~F (¯ x ) , ~G (¯ x )) = 0 , for k = 1 , · · · , k max , ¯ x ∈ (0 , δ ) ; and that [ µ l (¯ x )]( ~F (¯ x ) , ~G (¯ x )) is bounded on (0 , δ ) and tends to zero as ¯ x → . Then there exist δ ′ > and ~F = ( F , · · · , F j max ) , ~G = ( G , · · · , G i max ) semialgebraic and in C m (Ω closure δ ′ , R j max ) and C m (Ω closure δ ′ , R i max ) respectively, with J (0 , ~F = 0 , J (0 , ~G = 0 , G i = P j A ij F j + B i and j ¯ x ~F = ~F (¯ x ) , j ¯ x ~G = ~G (¯ x ) , for all ¯ x ∈ (0 , δ ′ ) . (Note that here we have passed from δ to a smaller δ ′ .) The remainder of this section is devoted to a proof of Lemma 6.2.Let δ >
Definition 6.1
We define a bundle H over [0 , × { } ⊂ R . Here, H = ( H (¯ x, ¯ x ∈ [0 , ,with H (¯ x, ⊂ P j max + i max defined as follows. • H (0 ,
0) = { } . • If ¯ x ∈ (0 , , then ( ~P , ~Q ) = ( P , · · · , P j max , Q , · · · , Q i max ) ∈ H (¯ x, if and only if y | α |− m ∂ α (X j A ij P j + B i − Q i ) (¯ x, y ) → as y → + , for each | α | ≤ m and each i . We will show that H is a bundle, i.e., H ( z ) is a translate of an R z -submodule of R j max + i max z for each z ∈ [0 , δ ] × { } ; and we will show that J (¯ x, ( ~F , ~G ) ∈ H (¯ x,
0) (each ¯ x ∈ [0 , δ ]) if ~F , ~G are as in (41).Suppose J (0 , ( ~F , ~G ) = 0, ~F , ~G are C m on Ω closure δ , G i = P j A ij F j + B i on Ω δ . Let¯ x ∈ (0 , δ ]. Then ∂ α [ A ij ( F j − J (¯ x, F j )](¯ x, y ) = o ( y m −| α | )and ∂ α [ G i − J (¯ x, G i ](¯ x, y ) = o ( y m −| α | )on Ω δ for | α | ≤ m , by Taylor’s theorem and our estimates for ∂ α A ij . The above remarksimply that ∂ α { P j A ij J (¯ x, F j + B i − J (¯ x, G i } (¯ x,
0) = o ( y m −| α | ).Therefore, J (¯ x, ( ~F , ~G ) ∈ H (¯ x,
0) for ¯ x ∈ (0 , δ ]. For ¯ x = 0, we just note that J (0 , ( ~F , ~G ) =0 ∈ H (0 , J (¯ x, ( ~F , ~G ).Note that for ¯ x = 0 , H (¯ x,
0) is a translate in P of I (¯ x ) = ((cid:16) ~P , ~Q (cid:17) : ∂ α X j A ij P i − Q i ! (¯ x, y ) = o (cid:0) y m −| α | (cid:1) , as y → + , | α | ≤ m ) .25et (cid:16) ~P , ~Q (cid:17) ∈ I (¯ x ) and let S ∈ P . Then for | α | ≤ m, we have ∂ α S · "X j A ij P j − Q i (¯ x, y ) = o (cid:0) y m −| α | (cid:1) , hence(43) ∂ α X j A ij ( SP j ) − ( SQ i ) ! (¯ x, y ) = o (cid:0) y m −| α | (cid:1) , as y → + .Also, our estimates on ∂ α A ij , together with Taylor’s theorem, give ∂ α (cid:0) A ij (cid:0) SP i − J (¯ x, ( SP j ) (cid:1)(cid:1) (¯ x,
0) = o (cid:0) y m −| α | (cid:1) and ∂ α (cid:0) SQ i − J (¯ x, ( SQ i ) (cid:1) (¯ x,
0) = o (cid:0) y m −| α | (cid:1) as y → + for | α | ≤ m .That is,(44) ∂ α (cid:0) A ij (cid:0) SP j − S ⊙ (¯ x, P j (cid:1)(cid:1) (¯ x, y ) = o (cid:0) y m −| α | (cid:1) and(45) ∂ α (cid:0) SQ i − S ⊙ (¯ x, Q i (cid:1) (¯ x,
0) = o (cid:0) y m −| α | (cid:1) as y → + for | α | ≤ m. It now follows from (43), (44), and (45) that ∂ α X j A ij (cid:2) S ⊙ (¯ x, P j (cid:3) − (cid:2) S ⊙ (¯ x, Q i (cid:3)! (¯ x, y ) = o (cid:0) y m −| α | (cid:1) as y → + , for each | α | ≤ m .This completes the proof that the I (¯ x ) is a submodule, when ¯ x = 0.For ¯ x = 0 , we just note that { } is an R (0 , -submodule of R j max + i max (0 , . We have now shown that • H = ( H (¯ x, ¯ x ∈ [0 ,δ ] is a bundle. • If ( ~F , ~G ) is as in (I) of Lemma 6.2, then ( ~F , ~G ) is a section of H . • H (¯ x, ⊂ P j max + i max depends semialgebraically on ¯ x , since A ij and B i are semialgebraic. Lemma 6.3
Let H = ( H (¯ x, (¯ x, ∈ [0 ,δ ] ×{ } be a semialgebraic bundle, H = H (¯ x, ⊂P j max + i max . Then there exist semialgebraic functions λ , · · · , λ k max : (0 , δ ) → Aff, and afinite set of bad points { ¯¯ x bad , · · · , ¯¯ x bad S } such that the following holds for any ¯¯ x ∈ (0 , δ ) other han the bad points. Let (cid:16) ~F , ~G (cid:17) = ( F , · · · , F j max , G , · · · , G i max ) be C m in a neighborhoodof (¯¯ x, in R . Then J (¯ x, (cid:16) ~F , ~G (cid:17) ∈ H (¯ x, for all ¯ x in some neighborhood of ¯¯ x if and only if [ λ k (¯ x )] (cid:16) j ¯ x ~F , j ¯ x ~G (cid:17) = 0 for all ¯ x in some neighborhood of ¯¯ x, ( k = 1 , · · · , k max ) . Proof.
This is a 1 dimensional case of Lemma 5.1, whose proof can be found in Section 5.1.
Proof of Lemma 6.2.
We apply Lemma 6.3 to the bundle H defined in Definition 6.1.By making δ smaller, we may assume there are no bad points ¯¯ x bad . Thus, we have achievedthe following: There exist semialgebraic functions λ , · · · , λ k max : (0 , δ ] → Aff such that forany ¯¯ x ∈ (0 , δ ) and any ( ~F , ~G ) that is C m in a neighborhood of (¯¯ x, J (¯ x, ( ~F , ~G ) ∈ H (¯ x,
0) for all ¯ x in some neighborhood of ¯¯ x if and only if[ λ k (¯ x )] j ¯ x (cid:16) ~F , ~G (cid:17) = 0 for all ¯ x in some neighborhood of ¯¯ x, ( k = 1 , · · · , k max ).In particular, if (cid:16) ~F , ~G (cid:17) is as in (41), then[ λ k (¯ x )] j ¯ x (cid:16) ~F , ~G (cid:17) = 0 for all ¯ x ∈ (0 , δ ) , ( k = 1 , · · · , k max ).Next, we apply Theorem 6 in Section 3.7.Recall H (¯ x,
0) is an affine space, so R · H (¯ x,
0) is a vector space.We regard R · H (¯ x,
0) as the space of all ( ~P , ~Q, t ) such that ∂ α { P j A ij P j + tB i − Q i } (¯ x, y ) = o ( y m −| α | ) as y → + .We define seminorms on R · H (¯ x,
0) by ||| ( ~P , ~Q, t ) ||| α,i,y = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y | α |− m ∂ α (X j A ij P j + tB i − Q i ) (¯ x, y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for fixed ¯ x and 0 < y < ψ (¯ x ). Notice that on H (¯ x, ||| ( ~P , ~Q ) ||| α,i,y = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y | α |− m ∂ α (X j A ij P j + B i − Q i ) (¯ x, y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for fixed ¯ x = 0 and 0 < y < ψ (¯ x ) , | α | ≤ m, i = 1 , · · · , i max .Note that sup α,i,y ||| ( ~P , ~Q ) ||| α,i,y
27s bounded for fixed (cid:16) ~P , ~Q (cid:17) ∈ H (¯ x, H (¯ x, x ∈ (0 , δ ) , there exist y σ ∈ (0 , ψ (¯ x ))( σ = 1 , · · · , σ max ) with σ max depending only on i max , j max , m such that for any (cid:16) ~P , ~Q (cid:17) ∈ H (¯ x, 0) for all ¯ x ∈ (0 , δ ) . By definition of H (¯ x, 0) and by the estimates ∂ α (cid:0) F j − J (¯ x, F j (cid:1) (¯ x, y ) = o (cid:0) y m −| α | (cid:1) ,∂ α (cid:16) G i − J (¯ x, G i (cid:17) (¯ x, y ) = o (cid:0) y m −| α | (cid:1) , and | ∂ α A ij ( x, y ) | ≤ Cy −| α | ,we therefore have the following:(62) For any ¯ x ∈ (0 , δ ) , any i = 1 , · · · , i max , and any | α | ≤ m , the quantity y | α |− m ∂ α (X j A ij F j + B i − G i ) (¯ x, y )is bounded as y varies over (0 , ψ (¯ x )) and tends to zero as y → + .We don’t yet know that the above convergence is uniform in ¯ x. Next, we recall from (42) the assumption that the µ l (¯ x ) (cid:16) ~F (¯ x ) , ~G (¯ x ) (cid:17) remain boundedas ¯ x varies over (0 , δ ) and moreover these quantities tend to zero as ¯ x → + .Thus, the quantities(63) ( y σ (¯ x )) a − m ∂ ay (X j A ij F j + B i − G i ) (¯ x, y σ (¯ x ))31or 0 ≤ a ≤ m, i = 1 , · · · , i max , σ = 1 , · · · , σ max , remain bounded as ¯ x varies over (0 , δ ), andtend to zero as ¯ x → + .Because those quantities are semialgebraic functions of one variable, we may pass to asmaller δ and assert for any b , say 0 ≤ b ≤ m , that(64) (cid:18) dd ¯ x (cid:19) b ( y σ (¯ x ) a − m ∂ ay "X j A ij F j + B i − G i (¯ x, y σ (¯ x )) ) = o (cid:0) ¯ x − b (cid:1) as ¯ x → + and this quantity is bounded for ¯ x bounded away from 0.For 0 ≤ a + b ≤ m, we will check that(65) (¯ x ) a + b − m (cid:18) dd ¯ x (cid:19) b ( ∂ ay "X j A ij F j + B i − G i (¯ x, y σ (¯ x )) ) = o (1)as ¯ x → + and the left-hand side is bounded.To see this, we write (cid:18) dd ¯ x (cid:19) b ( ∂ ay "X j A ij F j + B i − G i (¯ x, y σ (¯ x )) ) = (cid:18) dd ¯ x (cid:19) b ( ( y σ (¯ x )) m − a ( y σ (¯ x )) a − m ∂ ay "X j A ij F j + B i − G i (¯ x, y σ (¯ x )) ) = X b ′ + b ′′ = b coeff ( b ′ , b ′′ ) "(cid:18) dd ¯ x (cid:19) b ′ ( y σ (¯ x )) m − a ( † ) · "(cid:18) dd ¯ x (cid:19) b ′′ ( ( y σ (¯ x )) a − m ∂ ay "X j A ij F j + B i − G i (¯ x, y σ (¯ x )) ) ( ‡ ) .Since y σ (¯ x ) is given by a Puiseux series for ¯ x ∈ (0 , δ ) (small enough δ ),( † ) = O ( y σ (¯ x )) m − a · ¯ x − b ′ = O (cid:16) y σ (¯ x ) m − a − b ′ (cid:17) , because 0 < y σ (¯ x ) < ψ (¯ x ) ≤ ¯ x . By (64), ( ‡ ) is o (cid:0) ¯ x − b ′′ (cid:1) as ¯ x → + .So in fact, we get not only (65) but the stronger result(66) (cid:18) dd ¯ x (cid:19) b ( ∂ ay "X j A ij F j + B i − G i (¯ x, y σ (¯ x )) ) = o (cid:0) y σ (¯ x ) m − a · ¯ x − b (cid:1) as ¯ x → + ; the left-hand side is bounded. 32ntroduce the vector field X σ = ∂∂x + y ′ σ (¯ x ) ∂∂y on R . We have (cid:18) dd ¯ x (cid:19) b {F (¯ x, y σ (¯ x )) } = ( X σ ) b F (cid:12)(cid:12)(cid:12) (¯ x,y σ (¯ x )) for any F ∈ C bloc (cid:0) R (cid:1) .Therefore, (66) yields(67) (cid:0) X bσ ∂ ay (cid:1) "X j A ij F j + B i − G i (¯ x, y σ (¯ x )) = o (cid:0) y σ (¯ x ) m − a · ¯ x − b (cid:1) as ¯ x → + and the left-hand side is bounded for all ¯ x , for a + b ≤ m, σ = 1 , · · · , σ max , i = 1 , · · · , i max .This implies that(68) ( y σ (¯ x )) | α |− m ∂ α hP j A ij F j + B i − G i i (¯ x, y σ (¯ x )) is bounded on (0 , δ ) and tends to zeroas ¯ x → + , for | α | ≤ m, i = 1 , · · · , i max , σ = 1 , · · · , σ max .Let α = ( b, a ) , ∂ α = ∂ bx ∂ ay .We deduce (68) from (67) by induction on b . For b = 0 , (68) is the same as (67).Assume we know (68) for all b ′ < b. We prove (68) for the given b, using our inductionhypothesis for b ′ , together with (67).The quantity(69) X bσ ∂ ay (X j A ij F j + B i − G i ) (¯ x, y σ (¯ x ))is a sum of terms of the form(70) (cid:0) ∂ b x y σ (¯ x ) (cid:1) · · · · · (cid:0) ∂ b ν x y σ (¯ x ) (cid:1) · ∂ ¯ bx ∂ a + νy (X j A ij F j + B i − G i ) (¯ x, y σ (¯ x ))with b t ≥ t , b + · · · + b ν + ¯ b = b. Note ¯ b + ( a + ν ) = a + ¯ b + b + · · · + b ν − ( b − − · · · − ( b ν − ≤ a + b .We know that (69) = o (cid:16) y σ (¯ x ) m − a − b (cid:17) by (67).If ¯ b < b, then by our induction hypothesis, the term (70) is dominated by O Here again we use 0 34y Taylor’s theorem, (cid:12)(cid:12) ∂ α (cid:8) F j − J (¯ x, F j (cid:9) (¯ x, y ) (cid:12)(cid:12) ≤ Cy m −| α | E (¯ x ) for | α | ≤ m, (¯ x, y ) ∈ Ω δ .Recall that | ∂ α A ij (¯ x, y ) | ≤ Cy −| α | for | α | ≤ m and (¯ x, y ) ∈ Ω δ .Just as we estimated the functions F j above, we have from Taylor’s theorem that (cid:12)(cid:12)(cid:12) ∂ α n G i − J (¯ x, G i o (¯ x, y ) (cid:12)(cid:12)(cid:12) ≤ Cy m −| α | E (¯ x ) for | α | ≤ m, (¯ x, y ) ∈ Ω δ .Combining these estimates, we see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ α (X j A ij (cid:0) F j − J (¯ x, F j (cid:1) − (cid:16) G i − J (¯ x, G i (cid:17)) ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cy m −| α | E (¯ x ) for | α | ≤ m, (¯ x, y ) ∈ Ω δ .(75)Combining (71), (74), (75), we see that( y σ (¯ x )) | α |− m ∂ α (X j A ij (cid:2) J (¯ x, F j (cid:3) + B i − h J (¯ x, G i i) (¯ x, y σ (¯ x ))(76) is bounded on (0 , δ ) and tends to 0 as ¯ x tends to 0 + .Recall that (cid:16) J (¯ x, ~F , J (¯ x, ~G (cid:17) ∈ H (¯ x ) for all ¯ x ∈ (0 , δ ] (see (61)).The above results, together with the property (50) of the y σ (¯ x ) now tells us that(77) y | α |− m ∂ α nP j A ij (cid:0) J (¯ x, F j (cid:1) + B i − (cid:16) J (¯ x, G i (cid:17)o (¯ x, y ) is bounded on Ω δ and tends tozero as (¯ x, y ) ∈ Ω δ tends to zero.Together with (74), (75), this yields the following result(78) y | α |− m ∂ α nP j A ij F j + B i − G i o (¯ x, y ) is bounded on Ω δ and tends to zero as (¯ x, y ) ∈ Ω δ tends to zero. Here, i = 1 , · · · , i max and | α | ≤ m are arbitrary.From (62), we have(79) lim y → + y | α |− m ∂ α (cid:16)P j A ij F j + B i − G i (cid:17) ( x, y ) = 0 for each fixed x ∈ (0 , δ ).The functions A ij , F j , B i , G i are semialgebraic. Therefore, by Lemma 3.3, there exista positive integer K and a semialgebraic function of one variable A ( x ) such that(80) (cid:12)(cid:12)(cid:12) y | α |− m ∂ α (cid:16)P j A ij F j + B i − G i (cid:17) ( x, y ) (cid:12)(cid:12)(cid:12) ≤ A ( x ) · y K for all ( x, y ) ∈ Ω δ , | α | ≤ m, i =1 , · · · , i max . 35aking δ smaller, we may assume A ( x ) is C ∞ on (0 , δ ].Consequently, y | α |− m ∂ α (cid:16)P j A ij F j + B i − G i (cid:17) ( x, y ) tends to zero as y → + , uniformlyas x varies over ( ε, δ ) for any ε > G i = P j A ij F j + B i , we see that for | α | ≤ m, i = 1 , · · · , i max , (81) y | α |− m ∂ α n G i − G i o ( x, y ) → y → + uniformly for x in each interval ( ε, δ ).Recalling that G i belongs to C ∞ in a neighborhood of ( x, 0) (each x ∈ (0 , δ )), weconclude that the derivatives ∂ α G i ( x, y ) ( | α | ≤ m, i = 1 , · · · , i max ), initially defined onΩ δ = { ( x, y ) : 0 < x < δ, < y < ψ ( x ) } extend to continuous functions on(82) Ω ++ δ ≡ { ( x, y ) : 0 < x < δ, ≤ y < ψ ( x ) } . Next, recall that F js is C ∞ on (0 , δ ) and that we assume that | ∂ α A ij ( x, y ) | , | ∂ α B i ( x, y ) | ≤ Cy −| α | on(83) Ω + = { ( x, y ) : 0 < x < δ, < y ≤ ψ ( x ) } on which the functions ∂ α A ij , ∂ α B i are assumed to be continuous.We defined G i = X j A ij F j + B i = X j A ij ( x, y ) " m X s =0 F js ( x ) y s + B i ( x, y ) .The above remarks (and the fact that ψ ( x ) = 0 for x ∈ (0 , δ )) show that ∂ α G i extendsto a continuous function on Ω + (see (83)), for | α | ≤ m, i = 1 , · · · , i max .Combining our results for Ω + (see (83)) and for Ω ++ (see (82)), we see that ∂ α G i extendsto a continuous function on Ω closure δ \ { (0 , } for each i = 1 , · · · , i max , | α | ≤ m .Also, ∂ α F i is a continuous function on Ω closure δ \ { (0 , } because F i is C ∞ on (0 , δ ) × R .By (72), we have G is ( x ) = o ( x m − s ) (0 ≤ s ≤ m ) on (0 , δ ). Because G is is semialgebraic,it follows that after possibly reducing δ , we have (cid:18) ddx (cid:19) t G is ( x ) = o (cid:0) x m − s − t (cid:1) for 0 ≤ t ≤ m, ≤ s ≤ m, i = 1 , · · · , i max .36ecause G i ( x, y ) = P ms =0 G is ( x ) y s and 0 < y < ψ ( x ) ≤ x on Ω δ , we have on Ω δ that (cid:12)(cid:12)(cid:12) ∂ tx ∂ sy G i ( x, y ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X s = s coeff ( s, s ) · (cid:18) ddx (cid:19) t G is ( x ) · y s − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o m X s = s x m − s − t · y s − s ! = o m X s = s x m − s − t · x s − s ! = o (cid:0) x m − s − t (cid:1) on Ω δ for s, t ≤ m .In particular,(84) ∂ α G i ( x, y ) → x, y ) ∈ Ω δ tends to (0 , 0) for | α | ≤ m, i = 1 , · · · , i max .On the other hand, recalling the definition G i = P j A ij F j + B i , we see from (78) that ∂ α (cid:16) G i − G i (cid:17) ( x, y ) → x, y ) ∈ Ω δ tends to (0 , 0) for each | α | ≤ m . Together with(84), this shows that ∂ α G i ( x, y ) → x, y ) ∈ Ω δ tends to (0 , 0) for each | α | ≤ m .Next, recall from (72) that F js ( x ) = o ( x m − s ) for x ∈ (0 , δ ), j = 1 , · · · , j max , s = 0 , · · · , m .Because the F jk are semialgebraic functions of one variable, we conclude (after reducing δ ) that (cid:0) ddx (cid:1) t F js ( x ) = o ( x m − s − t ) on (0 , δ ) for t ≤ m .Now, for s + t ≤ m and ( x, y ) ∈ Ω δ (hence 0 < y < ψ ( x ) ≤ x ), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂∂y (cid:19) s (cid:18) ∂∂x (cid:19) t F j ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂∂y (cid:19) s (cid:18) ∂∂x (cid:19) t m X s =0 F js ( x ) y s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X s = s coeff ( s, s ) "(cid:18) ddx (cid:19) t F js ( x ) · y s − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C m X s = s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ddx (cid:19) t F js ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · x s − s = o m X s =0 x m − s − t x s − s ! = o (cid:0) x m − s − t (cid:1) .Thus, for | α | ≤ m , and j = 1 , · · · , j max , we have ∂ α F j ( x, y ) → x, y ) ∈ Ω δ tends to (0 , 0) .We now know the following: G i = P j A ij F j + B i on Ω δ . The F j and G i are semialgebraicon Ω δ For | α | ≤ m , the derivatives ∂ α F j , ∂ α G i extend to continuous functions on Ω closure2 δ/ \{ (0 , } . For | α | ≤ m, the derivatives ∂ α F j ( z ), ∂ α G i ( z ) tend to zero as z ∈ Ω δ tends to zero.37t follows that the F j and G i extend from Ω δ/ to semialgebraic functions in C m (cid:16) Ω closure δ/ (cid:17) and those functions all have m -jet zero at the origin. We extend F j , G i to semialgebraic C mloc functions on R , using Corollary 3.2.Next, we show that j ¯ x (cid:16) ~F , ~G (cid:17) = (cid:16) ~F (¯ x ) , ~G (¯ x ) (cid:17) for ¯ x ∈ (0 , δ ).From (60), we have j ¯ x (cid:16) ~F , ~G (cid:17) = (cid:16) ~F (¯ x ) , ~G (¯ x ) (cid:17) .From (81), we see that j ¯ x (cid:16) G i − G i (cid:17) = 0 for all ¯ x ∈ (0 , δ ). Therefore, j ¯ x (cid:16) ~F , ~G (cid:17) = j ¯ x (cid:16) ~F , ~G (cid:17) = (cid:16) ~F (¯ x ) , ~G (¯ x ) (cid:17) ,as desired.Thus, we have proven (42).The proof of Lemma 6.2 is complete. Lemma 6.4 Let ψ ( x ) be a semialgebraic function on [0 , δ ] , satisfying ψ (0) = 0 , < ψ ( x ) ≤ x for all x ∈ (0 , δ ] . We set E δ = { ( x, y ) ∈ R : 0 ≤ x ≤ δ, ≤ y ≤ ψ ( x ) } ,E + δ = { ( x, y ) ∈ R : 0 ≤ x ≤ δ, ψ ( x ) ≤ y ≤ ψ ( x ) } , and E − δ = { ( x, y ) ∈ R : 0 ≤ x ≤ δ, ≤ y ≤ ψ ( x ) } . Fix a semialgebraic function of one variable, θ ( t ) , satisfying ≤ θ ( t ) ≤ , θ ( t ) = 1 for t ≤ / , θ ( t ) = 0 for t ≥ / , θ ∈ C m +100 .Then set θ − ( x, y ) = θ (cid:18) yψ ( x ) (cid:19) , θ + ( x, y ) = 1 − θ − ( x, y ) for ( x, y ) ∈ E δ \ { (0 , } .Thus, θ + , θ − ≥ and θ + + θ − = 1 on E δ \ { (0 , } .Let F + ∈ C m ( E + δ ) and F − ∈ C m ( E − δ ) be semialgebraic functions, with J (0 , F + = J (0 , F − = 0 .Suppose that (85) ∂ ly F + ( x, ψ ( x )) − m − l X j =0 j ! ∂ l + jy F − ( x, · ( ψ ( x )) j = o (( ψ ( x )) m − l ) as x → + for each l = 0 , · · · , m .Define F = θ + · F + + θ − · F − on E δ \ { (0 , } , F (0 , 0) = 0 .Then F is a C m semialgebraic function on E δ ′ for some small δ ′ . The jet of F at theorigin is zero. Moreover, F = F + in a neighborhood of any point ( x, ψ ( x )) , < x < δ ′ ; and F = F − in a neighborhood of any point ( x, , < x < δ ′ . roof. Because 0 ≤ ψ ( x ) ≤ x and ψ is given near 0 by a convergent Puiseux series, wehave ψ ( k ) ( x ) = O ( x − k ) as x → + , for k = 0 , · · · , m + 100. Also, because F + , F − havezero jet at (0 , | α | = m , ∂ α F + ( x, y ) = o (1) as ( x, y ) ∈ E + δ tends to zero and ∂ α F − ( x, y ) = o (1) as ( x, y ) ∈ E − δ tends to zero.By induction on µ , we now prove that(86) ∂ µx ∂ ly F + ( x, ψ ( x )) − P m − l − µj =0 1 j ! ∂ µx ∂ l + jy F − ( x, · ( ψ ( x )) j = o (( ψ ( x )) m − µ − l ) as x → + for µ + l ≤ m .For µ = 0, (86) is a hypothesis of our lemma. Assuming (86) for µ , we prove it for µ + 1. Thus, fix l satisfying ( µ + 1) + l ≤ m . Recalling that ∂ µx ∂ l + jy F − ( x, 0) = o (1) when µ + ( l + j ) = m , we conclude from (86) that(87) ∂ µx ∂ ly F + ( x, ψ ( x )) − P m − l − µ − j =0 1 j ! ∂ µx ∂ l + jy F − ( x, · ( ψ ( x )) j = o (( ψ ( x )) m − µ − l ) as x → + .Because the above functions are semialgebraic functions of one variable and thus givennear 0 by convergent Puiseux series, it follows that ddx { (87) } = o (( ψ ( x )) m − µ − l · x − ), hence ddx { (87) } = o (( ψ ( x )) m − µ − l − ), because 0 < ψ ( x ) ≤ x . Thus, (cid:2) ( ∂ x + ψ ′ ( x ) ∂ y ) (cid:0) ∂ µx ∂ ly F + (cid:1)(cid:3) ( x, ψ ( x )) − m − l − µ − X j =0 j ! ∂ µ +1 x ∂ l + jy F − ( x, 0) ( ψ ( x )) j − m − l − µ − X j =1 j ! ∂ µx ∂ l + jy F − ( x, j ( ψ ( x )) j − ψ ′ ( x )= o (cid:16) ( ψ ( x )) m − µ − l − (cid:17) .It follows that ∂ µ +1 x ∂ ly F + ( x, ψ ( x )) − m − l − ( µ +1) X j =0 j ! ∂ µ +1 x ∂ l + jy F − ( x, 0) ( ψ ( x )) j + ψ ′ ( x ) " ∂ µx ∂ l +1 y F + ( x, ψ ( x )) − m − l − µ − X j =0 j ! ∂ µx ∂ l +1+ jy F − ( x, 0) ( ψ ( x )) j (88) = o (cid:16) ( ψ ( x )) m − ( µ +1) − l (cid:17) .For j = m − l − µ − 1, we have ∂ µx ∂ l +1+ jy F − ( x, 0) = o (1), hence inductive hypothesis(86) for ( l + 1) in place of l tells us that the second term in square brackets in (88) is o (cid:16) ( ψ ( x )) m − ( µ +1) − l (cid:17) . Also, | ψ ′ ( x ) | = O (1).Consequently, the first term in square brackets in (88) is o (cid:16) ( ψ ( x )) m − ( µ +1) − l (cid:17) , provingthe analogue of (86) for µ + 1 , thus completing the induction and establishing (86) . 39e bring in the cutoff functions θ + and θ − . Note that θ + is supported in E + δ and θ − issupported in E − δ .We will estimate the derivatives of θ + , θ − on E δ .We have (cid:18) ddx (cid:19) k ψ ( x ) = O (cid:18) ψ ( x ) x − k (cid:19) as x → + , because ψ is given by a convergent Puiseux series.Because 0 < ψ ( x ) ≤ x for x ∈ (0 , δ ) and 0 ≤ y ≤ ψ ( x ) in E δ , it follows that ∂ lx ∂ ky (cid:18) yψ ( x ) (cid:19) = O (cid:16) ( ψ ( x )) − k − l (cid:17) as ( x, y ) ∈ E δ → 0, for all k, l ≥ ∂ αx,y θ − ( x, y ) is a sum of terms θ ( s ) (cid:16) yψ ( x ) (cid:17) · Q sσ =1 h ∂ α σ x,y (cid:16) yψ ( x ) (cid:17)i with α + · · · + α s = α , s ≤ | α | .Each such term is O (cid:18)Q sσ =1 (cid:16) ψ ( x ) (cid:17) | α σ | (cid:19) = O (cid:18)(cid:16) ψ ( x ) (cid:17) | α | (cid:19) .Thus,(89) (cid:12)(cid:12) ∂ αx,y θ − ( x, y ) (cid:12)(cid:12) , (cid:12)(cid:12) ∂ αx,y θ + ( x, y ) (cid:12)(cid:12) ≤ C α ( ψ ( x )) | α | on E δ (smaller δ ) for | α | ≤ m + 100.Next, we return to F + , F − , and prove the following estimate(90) ∂ µx ∂ ly (cid:0) F + − F − (cid:1) ( x, y ) = o (cid:16) [ ψ ( x )] m − µ − l (cid:17) as ( x, y ) ∈ E + δ ∩ E − δ → µ, l with µ + l ≤ m .To see this, fix µ , 0 ≤ µ ≤ m , and look at the polynomials P + x ( y ) = m − µ X j =0 j ! (cid:2) ∂ jy ∂ µx F + ( x, ψ ( x )) (cid:3) · ( y − ψ ( x )) j , P − x ( y ) = m − µ X j =0 j ! (cid:2) ∂ jy ∂ µx F − ( x, (cid:3) · y j .Estimate (86) shows that(91) ∂ ly (cid:0) P + x − P − x (cid:1) | y = ψ ( x ) = o (cid:16) ( ψ ( x )) m − µ − l (cid:17) for l = 0 , · · · , m − µ .For y satisfying ( x, y ) ∈ E + δ ∩ E − δ , we have | y | , | y − ψ ( x ) | ≤ ψ ( x ) and therefore (91) yields ∂ ly (cid:0) P + x − P − x (cid:1) ( x, y ) = o (cid:16) ( ψ ( x )) m − µ − l (cid:17) as ( x, y ) ∈ E + δ ∩ E − δ tends to zero. 40n the other hand, Taylor’s theorem gives for ( x, y ) ∈ E + δ ∩ E − δ \ { (0 , } the estimates ∂ ly (cid:2) ∂ µx F + − P + x (cid:3) ( x, y ) = O ( ψ ( x )) m − µ − l · max ¯ y ∈ [ ψ ( x ) ,ψ ( x ) ] (cid:12)(cid:12) ∂ m − µy ∂ µx F + ( x, ¯ y ) (cid:12)(cid:12)! and ∂ ly (cid:2) ∂ µx F − − P − x (cid:3) ( x, y ) = O ( ψ ( x )) m − µ − l · max ¯ y ∈ [ , ψ ( x ) ] (cid:12)(cid:12) ∂ m − µy ∂ µx F − ( x, ¯ y ) (cid:12)(cid:12)! .The maxima in these last two estimates are o (1), because J (0 , F + = J (0 , F − = 0.Thus, as ( x, y ) ∈ E + δ ∩ E − δ \{ (0 , } approaches zero, the quantities ∂ ly [ ∂ µx F + − P + x ] ( x, y ), ∂ ly [ ∂ µx F − − P − x ] ( x, y ), ∂ ly [ P + x − P − x ] ( x, y ) are all o (cid:16) ( ψ ( x )) m − µ − l (cid:17) .Consequently, (cid:0) ∂ ly ∂ µx F + − ∂ ly ∂ µx F − (cid:1) ( x, y ) = o (cid:16) ( ψ ( x )) m − µ − l (cid:17) as ( x, y ) ∈ E + δ ∩ E − δ \{ (0 , } approaches zero, completing the proof of (90).We now set F = θ + F + + θ − F − on E δ \ { (0 , } and F (0 , 0) = 0.Evidently, F is C m away from the origin, and semialgebraic; moreover, F = F + in aneighborhood of any point ( x , ψ ( x )) in E δ ( x = 0) and F = F − in a neighborhood of anypoint ( x , ∈ E δ ( x = 0).It remains to check that F ∈ C m ( E δ ) near 0 and that J (0 , F = 0. That amounts toshowing that(92) ∂ αx,y F ( x, y ) = o (cid:0) x m −| α | (cid:1) as ( x, y ) ∈ E δ \ { (0 , } approaches (0 , 0) (all | α | ≤ m ).To prove (92), we may assume ( x, y ) ∈ E + δ ∩ E − δ \{ (0 , } , because otherwise the left-handside of (92) is ∂ αx,y F + for ( x, y ) ∈ E + δ \ { (0 , } or else ∂ αx,y F − for ( x, y ) ∈ E − δ \ { (0 , } , inwhich case (92) holds because J (0 , F + = J (0 , F − = 0.For ( x, y ) ∈ E + δ ∩ E − δ \ { (0 , } , we have(93) F = F − + θ + (cid:0) F + − F − (cid:1) .Because J (0 , F − = 0, we have(94) ∂ αx,y F − ( x, y ) = o (cid:0) x m −| α | (cid:1) as ( x, y ) ∈ E + δ ∩ E − δ \ { (0 , } tends to (0 , | α | ≤ m .We recall that ∂ αx,y θ + ( x, y ) = O (cid:16) ( ψ ( x )) −| α | (cid:17) for | α | ≤ m and that ∂ αx,y ( F + − F − ) ( x, y ) = o (cid:16) ( ψ ( x )) m −| α | (cid:17) for | α | ≤ m as ( x, y ) ∈ E + δ ∩ E − δ \ { (0 , } tends to (0 , | α | ≤ m .Therefore, for | α | ≤ m , as ( x, y ) ∈ E + δ ∩ E − δ \ { (0 , } tends to (0 , ∂ αx,y (cid:8) θ + (cid:0) F + − F − (cid:1) ( x, y ) (cid:9) = o (cid:16) ( ψ ( x )) m −| α | (cid:17) ,hence(95) ∂ αx,y (cid:8) θ + (cid:0) F + − F − (cid:1) ( x, y ) (cid:9) = o (cid:0) x m −| α | (cid:1) ,41ecause 0 < ψ ( x ) ≤ x . Putting (94), (95) into (93), we see that ∂ αx,y F ( x, y ) = o (cid:0) x m −| α | (cid:1) as ( x, y ) ∈ E + δ ∩ E − δ \ { (0 , } tends to (0 , | α | ≤ m .Thus, (92) holds. The proof of Lemma 6.4 is complete.Next, we introduce a change of variables in a neighborhood of 0 in R = { ( x, y ) : x > } of the form(96) ¯ x = x, ¯ y = y + ˜ ψ ( x ) ,where ˜ ψ ( x ) is semialgebraic and satisfies (cid:12)(cid:12)(cid:12) ˜ ψ ( x ) (cid:12)(cid:12)(cid:12) ≤ Cx for x ∈ (0 , δ ).The inverse change of variables is of course x = ¯ x, y = ¯ y − ˜ ψ (¯ x ) .Note that ∂ αx,y (¯ x, ¯ y ) = O (cid:0) x −| α | (cid:1) for | y | ≤ Cx ≪ ψ is given near 0 as a convergentPuiseux series, hence (cid:12)(cid:12)(cid:12) ˜ ψ ( x ) (cid:12)(cid:12)(cid:12) ≤ Cx implies (cid:12)(cid:12)(cid:12) ˜ ψ ( k ) (cid:12)(cid:12)(cid:12) ≤ C k x − k for small x .The change of variables (96) does not preserve C m , but it does preserve C m functionswhose jets at 0 are equal to zero.Indeed, suppose F (¯ x, ¯ y ) ∈ C m (cid:0) ¯ E (cid:1) for ¯ E ⊂ { (¯ x, ¯ y ) : | ¯ y | ≤ C ¯ x } , with 0 ∈ ¯ E and J F = 0.Then ¯ E corresponds under (96) to a set E ⊂ { ( x, y ) : | y | ≤ C ′ x } , 0 ∈ E .We may regard F as a function of ( x, y ), and for | α | ≤ m , ∂ αx,y F ( x, y ) is a sum of terms (cid:12)(cid:12)(cid:12) ∂ β ¯ x, ¯ y F (¯ x, ¯ y ) (cid:12)(cid:12)(cid:12) · Q | β | ν =1 (cid:2) ∂ α ν x,y (¯ x, ¯ y ) (cid:3) with | β | ≤ m and P ν α ν = α . If J (0 , F = 0 as a function of(¯ x, ¯ y ), then ∂ β ¯ x, ¯ y F (¯ x, ¯ y ) = o (cid:0) ¯ x m −| β | (cid:1) on ¯ E , hence ∂ β ¯ x, ¯ y F (¯ x, ¯ y ) = o (cid:0) x m −| β | (cid:1) on E . Also, on E, | β | Y ν =1 (cid:2) ∂ α ν x,y (¯ x, ¯ y ) (cid:3) = | β | Y ν =1 O (cid:0) x −| α ν | (cid:1) = O (cid:0) x | β |− P ν | α ν | (cid:1) = O (cid:0) x | β |−| α | (cid:1) .Consequently, ∂ αx,y F ( x, y ) = o (cid:0) x m −| α | (cid:1) on E \ { (0 , } , for | α | ≤ m . Thus, as claimed, F ∈ C m ( E ) and J (0 , F = 0.The following generalization of Lemma 6.4 is reduced to Lemma 6.4 by means of thechange of variables discussed above. Lemma 6.5 Let ≤ ψ − ( x ) ≤ ψ + ( x ) ≤ x be semialgebraic functions on [0 , δ ] , with ψ − < ψ + on (0 , δ ] . We set E δ = { ( x, y ) ∈ R : 0 ≤ x ≤ δ, ψ − ( x ) ≤ y ≤ ψ + ( x ) } ,E + δ = { ( x, y ) ∈ R : 0 ≤ x ≤ δ, ≤ ψ + ( x ) − y ≤ 23 ( ψ + ( x ) − ψ − ( x )) } , and E − δ = { ( x, y ) ∈ R : 0 ≤ x ≤ δ, ≤ y − ψ − ( x ) ≤ 23 ( ψ + ( x ) − ψ − ( x )) } . ix a semialgebraic function of one variable, θ ( t ) , satisfying ≤ θ ( t ) ≤ , θ ( t ) = 1 for t ≤ / , θ ( t ) = 0 for t ≥ / , θ ∈ C m +100 .Then set θ − ( x, y ) = θ (cid:18) y − ψ − ( x )( ψ + − ψ − ) ( x ) (cid:19) , θ + ( x, y ) = 1 − θ − ( x, y ) for ( x, y ) ∈ E δ \ { (0 , } .Thus, θ + , θ − ≥ and θ + + θ − = 1 on E δ \ { (0 , } .Let F + ∈ C m ( E + δ ) and F − ∈ C m ( E − δ ) be semialgebraic functions, with J (0 , F + = J (0 , F − = 0 .Suppose that ∂ ly F + ( x, ψ + ( x )) − m − l X j =0 j ! ∂ l + jy F − ( x, ψ − ( x )) · ( ψ + ( x ) − ψ − ( x )) j = o (( ψ + ( x ) − ψ − ( x )) m − l ) as x → + for each l = 0 , · · · , m .Define F = θ + · F + + θ − · F − on E δ \ { (0 , } , F (0 , 0) = 0 .Then F is a C m semialgebraic function on E δ ′ for some small δ ′ . The jet of F at (0 , iszero. Moreover, F = F + in a neighborhood of any point ( x, ψ + ( x )) , < x < δ ′ , and F = F − in a neighborhood of any point ( x, ψ − ( x )) , < x < δ ′ . Let H = ( H ( z )) z ∈ R be a semialgebraic bundle with a C mloc section. Each H ( z ) is a coset ofan R z submodule in R Dz . Assume H ((0 , { } . Let Ω δ = { ( x, y ) ∈ R : 0 ≤ x ≤ δ, ≤ y ≤ x } for δ > 0. We look for semialgebraic C mloc sections of H| Ω δ , for some small δ (whichwill keep shrinking as we discuss further).We apply Lemma 5.3. Thus, we obtain the following • Semialgebraic functions 0 ≤ ψ ( x ) ≤ ψ ( x ) ≤ · · · ≤ ψ s max ( x ) = x on (0 , δ ) , all givenby convergent Puiseux expansions on (0 , δ ). • Integers k s (0 ≤ k s ≤ D ) and permutations π s : { , · · · , D } → { , · · · , D } for s =1 , · · · , s max . • Semialgebraic functions A sij ( x, y ) ( s = 1 , · · · , s max , 1 ≤ i ≤ k s , k s < j ≤ D ) and ϕ si ( x, y )( s = 1 , · · · , s max , ≤ i ≤ k s ) defined on E s = { ( x, y ) : 0 < x < δ, ψ s − ( x ) < y < ψ s ( x ) } . • Semialgebraic functions θ sijl ( x ), g si ( x ) ( s = 0 , · · · , s max , i = 1 , · · · , i max ( s ), j = 1 , · · · , D,l = 0 , · · · , m ) defined on (0 , δ ), and given there by convergent Puiseux expansions.The above objects have the following properties • (Estimates) For ( x, y ) ∈ Ω with 0 < x < δ and ψ s − ( x ) < y < ψ s ( x ), we have (cid:12)(cid:12) ∂ α A sij ( x, y ) (cid:12)(cid:12) , | ∂ α ϕ si ( x, y ) | ≤ C [min ( | y − ψ s ( x ) | , | y − ψ s − ( x ) | )] −| α | for | α | ≤ m + 100.43 (Condition for sections) Let F = ( F , · · · , F D ) ∈ C m (cid:0) Ω , R D (cid:1) , and suppose J x F ∈ H ( x ) for all x ∈ Ω .Then for s = 1 , · · · , s max , i = 1 , · · · , k s , x ∈ (0 , δ ), ψ s − ( x ) < y < ψ s ( x ), we have(97) F π s i ( x, y ) + X D ≥ j>k s A sij ( x, y ) F π s j ( x, y ) = ϕ si ( x, y ) ;and for s = 0 , , · · · , s max , i = 1 , · · · , i max ( s ), x ∈ (0 , δ ), we have(98) D X j =1 m X l =0 θ sijl ( x ) ∂ ly F j ( x, ψ s ( x )) = g si ( x ) ;and(99) J (0 , F j = 0for all j .Conversely, if F = ( F j ) j =1 , ··· ,D ∈ C mloc (cid:0) R , R D (cid:1) satisfies (97), (98), (99), then F is asection of H over Ω closure δ .Next, we set (for s = 1 , · · · , s max ): E + s = (cid:26) ( x, y ) ∈ R : 0 ≤ x ≤ δ, ≤ ψ s ( x ) − y ≤ 23 ( ψ s − ψ s − ( x )) (cid:27) and E − s = (cid:26) ( x, y ) ∈ R : 0 ≤ x ≤ δ , 0 ≤ y − ψ s − ( x ) ≤ 23 ( ψ s ( x ) − ψ s − ( x )) (cid:27) .Then E + , interior s ∪ E − , interior s = E s . On E + , interior s we have (cid:12)(cid:12) ∂ α A sij ( x ) (cid:12)(cid:12) , | ∂ α ϕ si ( x, y ) | ≤ C ( ψ s ( x ) − y ) −| α | for | α | ≤ m + 100, and on E − , interior s we have (cid:12)(cid:12) ∂ α A sij ( x ) (cid:12)(cid:12) , | ∂ α ϕ si ( x, y ) | ≤ C ( y − ψ s − ( x )) −| α | for | α | ≤ m + 100.We may apply Lemma 6.2 after a change of variables of the form (¯ x, ¯ y ) = ( x, ± ( y − ψ ( x ))) . Thus, we obtain the following objects, with properties described below. • Semialgebraic functions θ + ,sijl ( x ), g + ,si ( x ), i = 1 , · · · , i +max ( s ), θ − ,sijl ( x ), g − ,si ( x ), i =1 , · · · , i − max ( s ), l = 0 , · · · , m, defined on (0 , δ ) (smaller δ ). • Semialgebraic functions ˜ θ + ,sijl ( x ), ˜ g + ,si ( x ), i = 1 , · · · , ˜ ı +max ( s ), ˜ θ − ,sijl ( x ), ˜ g − ,si ( x ), i =1 , · · · , ˜ ı − max ( s ), l = 0 , · · · , m, defined on (0 , δ ) (smaller δ ).The properties for these functions are as follows.Let F = ( F , · · · , F D ) ∈ C mloc (cid:0) R , R D (cid:1) satisfy (97) in E + , interior s and J (0 , F = 0. Then(100) X ≤ j ≤ D ≤ l ≤ m θ + ,sijl ∂ ly F j ( x, ψ s ( x )) = g + ,si ( x )44or x ∈ (0 , δ ) and all i , and(101) X ≤ j ≤ D ≤ l ≤ m ˜ θ + ,sijl ∂ ly F j ( x, ψ s ( x )) = ˜ g + ,si ( x ) + o (1) as x → + for x ∈ (0 , δ ) and all i .Similarly, let F = ( F , · · · , F D ) ∈ C mloc (cid:0) R , R D (cid:1) satisfy (97) in E − , interior s and J (0 , F = 0.Then(102) X ≤ j ≤ D ≤ l ≤ m θ − ,sijl ∂ ly F j ( x, ψ s − ( x )) = g − ,si ( x )for x ∈ (0 , δ ) and all i , and(103) X ≤ j ≤ D ≤ l ≤ m ˜ θ − ,sijl ∂ ly F j ( x, ψ s − ( x )) = ˜ g − ,si ( x ) + o (1) as x → + for all i .(104) Conversely, fix s and suppose we are given semialgebraic functions f + ,sjl ( x ) on (0 , δ )satisfying X ≤ j ≤ D ≤ l ≤ m θ + ,sijl f + ,sjl ( x ) = g + ,si ( x ) (all i )and X ≤ j ≤ D ≤ l ≤ m ˜ θ + ,sijl f + ,sjl ( x ) = ˜ g + ,si ( x ) + o (1) as x → + (all i ) . Then there exists a semialgebraic function F = ( F , · · · , F D ) ∈ C m (cid:0) E + s , R D (cid:1) such that(97) holds in E + , interior s and ∂ ly F j ( x, ψ s ( x )) = f + ,sjl ( x ) and J (0 , F j = 0 for all j .(105) Similarly, fix s and suppose we are given we are given semialgebraic functions f − ,sjl ( x )on (0 , δ ) satisfying X ≤ j ≤ D ≤ l ≤ m θ − ,sijl f − ,sjl ( x ) = g − ,si ( x ) (all i )and X ≤ j ≤ D ≤ l ≤ m ˜ θ − ,sijl f − ,sjl ( x ) = ˜ g − ,si ( x ) + o (1) as x → + (all i ) . Then there exists a semialgebraic function F = ( F , · · · , F D ) ∈ C m (cid:0) E − s , R D (cid:1) such that(97) holds in E − , interior s and ∂ ly F j ( x, ψ s ( x )) = f − ,sjl ( x ) and J (0 , F j = 0 for all j .45106) Moreover, if F = ( F , · · · , F D ) ∈ C m (cid:0) E closure s , R D (cid:1) with J (0 , F = 0, then f + ,sjl = ∂ ly F j ( x, ψ s ( x )) and f − ,sjl = ∂ ly F j ( x, ψ s − ( x )) satisfy the key hypothesis of Lemma 6.5,namely, f + ,sjl ( x ) − m − l X k =0 k ! f − ,sj ( l + k ) ( x ) ( ψ s ( x ) − ψ s − ( x )) k = o (cid:16) [ ψ s ( x ) − ψ s − ( x )] m − l (cid:17) as x → + by Taylor’s theorem.Now, suppose F = ( F , · · · , F D ) ∈ C mloc (cid:0) R , R D (cid:1) is a section of H over Ω δ . Then, setting f sjl ( x ) = ∂ ly F j ( x, ψ s ( x )) for x ∈ (0 , δ ) (smaller δ ), we learn that (because the F j satisfy (97),(98), (99)), properties (98) · · · (103) yield a collection of assertions of the form(107) X j =1 , ··· ,Dl =0 , ··· ,m θ ,sijl ( x ) f sjl ( x ) = g ,si ( x ) on (0 , δ )and(108) X j =1 , ··· ,Dl =0 , ··· ,m ˜ θ ,sijl ( x ) f sjl ( x ) = ˜ g ,si ( x ) + o (1) as x → + ;and also from (106) we have(109) f sjl ( x ) = m − l X k =0 k ! f s − j ( l + k ) ( x ) [ ψ s ( x ) − ψ s − ( x )] k + o (cid:16) [ ψ s ( x ) − ψ s − ( x )] m − l (cid:17) as x → + .Conversely, if the f sjl ( x ) are semialgebraic functions of one variable, satisfying (107),(108), and (109), then for each s = 1 , · · · , s max there exist F s + = ( F s + , , · · · , F s + ,D ) ∈ C m (cid:16) E s , closure+ , R D (cid:17) , F s − = ( F s − , , · · · , F s − ,D ) ∈ C m (cid:16) E s , closure − , R D (cid:17) semialgebraic such that(97), (98), (99) hold in E + s , E − s , respectively and ∂ ly F s + ,j ( x, ψ s ( x )) = f sjl ( x ), ∂ ly F s − ,j ( x, ψ s − ( x )) = f s − jl ( x ) and J (0 , F s + = J (0 , F s − = 0.Note that F s + is a section of H over E + s , and F s − is a section of H over E − s .Thanks to (109) and Lemma 6.5, we may patch together F s + , F s − into a semialgebraic F s = ( F s, , · · · , F s,D ) ∈ C m ( E closure s , R D ) such that J (0 , F s = 0, F s is a section of H over E closure s , and ∂ ly F sj ( x, ψ ( x )) = f sjl ( x ) and ∂ ly F sj ( x, ψ s − ( x )) = f s − jl ( x ).Because of these conditions, the F s ( s = 1 , · · · , s max ) fit together (their transverse deriva-tives up to order m match at the boundaries where the E s meet), so using also Corollary3.2, we obtain from the F s a single semialgebraic F = ( F , · · · , F D ) ∈ C mloc ( R , R D ) such that J (0 , F = 0, and F is a section of H over Ω δ .Thus, we have proven Lemma 6.1. 46 Proof of Lemma 4.1 (Main Lemma) From the Second Main Lemma (Lemma 6.1), we can easily deduce Lemma 4.1.Indeed, suppose H = ( H ( x, y )) ( x,y ) ∈ Ω δ is as in the hypotheses of Lemma 4.1.Let θ sijl , g si , ˜ θ sijl , ˜ g si , ψ s be as in Lemma 6.1.For x ∈ (0 , δ ) with δ small enough, we introduce the following objects: W ( x ) = (cid:0) ξ sjl (cid:1) ≤ s ≤ s max ≤ l ≤ m ≤ j ≤ D ∈ R ( s max +1) · ( m +1) · D : X j,l θ sijl ( x ) ξ sjl = g si ( x ) , each s, i , F (cid:0)(cid:0) ξ sjl (cid:1) , x (cid:1) = X s,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j,l ˜ θ sijl ( x ) ξ sjl − ˜ g si ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X s =0 X j,l (cid:12)(cid:12)(cid:12) ξ sjl − P m − lk =0 1 k ! ξ s − j ( l + k ) · ( ψ s ( x ) − ψ s − ( x )) k (cid:12)(cid:12)(cid:12) [ ψ s ( x ) − ψ s − ( x )] m − l , F min ( x ) = inf (cid:8) F (cid:0)(cid:0) ξ sjl (cid:1) , x (cid:1) : (cid:0) ξ sjl (cid:1) ∈ W ( x ) (cid:9) , andΞ OK ( x ) = (cid:8)(cid:0) ξ sjl (cid:1) ∈ W ( x ) : F (cid:0)(cid:0) ξ sjl (cid:1) , x (cid:1) ≤ F min ( x ) + x (cid:9) .Because θ sijl , g si , ˜ θ sijl , ˜ g si , ψ s are semialgebraic, the objects defined above depend semialge-braically on x . Thanks to conclusion (39) of Lemma 6.1, each W ( x ) and each Ξ OK ( x ) isnon-empty, and(110) F min ( x ) → x → + .From Theorem 3 we obtain(111) Semialgebraic functions ξ sjl ( x ) on (0 , δ ) such that (cid:0) ξ sjl ( x ) (cid:1) ∈ Ξ OK ( x ) for each x ∈ (0 , δ ).In particular, for x ∈ (0 , δ ), we have X j,l θ s,ijl ( x ) ξ sjl ( x ) = g si ( x ) for each s, i, j ;(112) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j,l ˜ θ sijl ( x ) ξ sjl ( x ) − ˜ g si ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [ F min ( x ) + x ] for each s, i ;(113)and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ sjl ( x ) − m − l X k =0 k ! ξ s − j ( l + k ) ( x ) · ( ψ s ( x ) − ψ s − ( x )) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [ F min ( x ) + x ] · ( ψ s ( x ) − ψ s − ( x )) m − l , for each s, j, l ( s = 0) .(114) 47rom (110), (113), (114), we see that(115) X j,l ˜ θ sijl ( x ) ξ sjl ( x ) = ˜ g si ( x ) + o (1) as x → + ,and ξ sjl ( x ) − m − l X k =0 k ! ξ s − j ( l + k ) ( x ) · ( ψ s ( x ) − ψ s − ( x )) k = o (cid:16) [ ψ s ( x ) − ψ s − ( x )] m − l (cid:17) as x → + .(116)Finally, from (111), (112), (115), (116), and the assertion (40) in Lemma 6.1, we concludethat H| Ω δ ′ has a C mloc semialgebraic section for some δ ′ < δ .This completes the proof of Lemma 4.1 and that of Theorem 1. 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