Smooth multiparameter perturbation of polynomials and operators

ESI The Erwin Schrödinger International Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien, Austria Smooth Multiparameter Perturbation of Polynomials and Operators Mark Losik Peter W. Michor Armin Rainer Vienna, Preprint ESI 1922 (2007) Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at May 22, 2007 SMOOTH MULTIPARAMETER PERTURBATION OF POLYNOMIALS AND OPERATORS MARK LOSIK, PETER W. MICHOR, ARMIN RAINER P j n−j be a family of polynoAbstract. Let P (x)(z) = zn + n j=1 (−1) aj (x)z mials of fixed degree n whose coefficients aj are germs at 0 of smooth (C ∞ ) complex valued functions defined near 0 ∈ Rq . We show: If P is generic there S exists a finite collection T of transformations Ψ : Rq , 0 → Rq , 0 such that {im(Ψ) : Ψ ∈ T } is a neighborhood of 0 and, for each Ψ ∈ T , the family P ◦ Ψ allows smooth parameterizations of its roots near 0. Any Ψ ∈ T is a finite composition of linear coordinate changes and transformations of the two types (x1 , . . . , xq ) 7→ (xi x1 , . . . , xi xi−1 , xi , xi xi+1 , . . . , xi xq ) and (x1 , . . . , xq ) 7→ (x1 , . . . , xi−1 , ±xN i , xi+1 , . . . , xq ), N ∈ N. As a consequence we prove that locally there exist parameterizations λi of the roots of generic P such that the gradient ∇λi ∈ L1 . This conclusion is best possible, i.e., in general the roots cannot be chosen with first partial derivatives in Lp for any 1 < p ≤ ∞. If P is a generic smooth family of hyperbolic polynomials (i.e., x 7→ P (x) has only real roots for each x) in a connected smooth manifold M , then there exists a smooth manifold M ′ and a surjective smooth projection Φ : M ′ → M which is a locally finite composition of blow-ups centered at single points, such that P ◦ Φ allows locally smooth roots on M ′ . By lifting to the universal covering π : M̃ ′ → M ′ we obtain global smooth roots of P ◦ Φ ◦ π on M̃ ′ . Our method allows to give a simple proof of Bronshtein’s theorem (under slightly stronger conditions): For hyperbolic (not necessarily generic) P with aj ∈ C n(n+1)/2 any continuous arrangement of its roots is locally Lipschitz. We give applications to the perturbation theory of normal (resp. Hermitian) matrices and unbounded normal (resp. selfadjoint) operators with compact resolvents and common domain of definition. 1. Introduction Let us consider a family of monic polynomials n P (x)(z) = z + n X (−1)j aj (x)z n−j j=1 where the coefficients aj : U → C (for 1 ≤ j ≤ n) are complex valued functions defined in an open subset U ⊆ Rq . If the coefficients aj are regular (of some kind) it is natural to ask whether the roots of P can be arranged regularly as well, i.e., whether it is possible to find n regular functions λj : U → C (for 1 ≤ j ≤ n) such that λ1 (x), . . . , λn (x) represent the roots of z 7→ P (x)(z) for each x ∈ U . This perturbation problem has been intensively studied under the following additional assumptions: (1) q = 1. (2) The polynomials P (x) are hyperbolic, i.e., all roots of P (x) are real. Date: May 25, 2007. 2000 Mathematics Subject Classification. 26C10, 30C15, 47A55, 47A56. Key words and phrases. multiparameter perturbation theory, smooth roots of polynomials. P.W.M. and A.R. were supported by ‘Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 17108-N04’. 1 2 M. LOSIK, P.W. MICHOR, A. RAINER If both of these conditions are satisfied, there exist real analytic parameterizations of the roots of P if its coefficients are real analytic, by a classical theorem due to Rellich [25]. If all aj are smooth (C ∞ ) and no two of the increasingly ordered continuous roots meet of infinite order of flatness, then there exist smooth parameterizations of the roots, see [1]. The roots may always be chosen twice differentiable provided that the aj are in C 3n, see [13]. The conclusion in this statement is best possible as shown by an example in [4]. If the polynomials P (x) are hyperbolic and all aj are in C n , but q > 1, then the roots of P may still be chosen locally in a Lipschitz way, which follows from the fundamental results of Bronshtein [5] and (alternatively) Wakabayashi [27]. A different and easier proof for the partial case that the coefficients aj are real analytic was recently given by Kurdyka and Paunescu [17]. We will also present a simple proof based on the method developed in this paper for C n(n+1)/2 coefficients in section 6. Furthermore, in [17] it is shown that, if the coefficients aj are real analytic, there exists a modification Φ : W → U , namely a locally finite composition of blow-ups with smooth centers, such that the roots of P ◦ Φ can be locally parameterized as real analytic functions. For further results on the perturbation problem of hyperbolic polynomials see [11], [9], [19], [6], [18]. The case when q = 1, but the hyperbolicity assumption is dropped, has been treated in [23]. In the one parameter case continuous parameterizations of the roots always exist if the coefficients aj are continuous (e.g. [12, II 5.2]). If all aj are smooth and no two of the continuously chosen roots meet of infinite order of flatness, then any continuous parameterization of the roots is locally absolutely continuous, see [23, 4.4]. The conclusion is best possible: the roots of P (x)(z) = z 2 − x (for x ∈ R) cannot fulfill a local Lipschitz condition near x = 0 (see also 5.4(1)). This theorem follows from the proposition that for any x0 there exists an integer N such that x 7→ P (x0 ± (x − x0 )N ) admits smooth parameterizations of its roots near x0 . Consider also Spagnolo [26]. In the present paper we study smooth multiparameter perturbations for complex polynomials, i.e., without the restrictions (1) and (2). By a theorem due to Ostrowski [20], for a continuous family P of polynomials, the set of all roots still is continuous and satisfies a Hölder condition of order 1/n. But in general there may not exist continuous parameterizations of single roots as in the one dimen2 sional or hyperbolic case. √ Take, for instance, P (x1 , x2)(z) = z − (x1 + ix2 ), with x1 , x2 ∈ R and P i = −1. Nevertheless, we show that for generic polynomials n P (x)(z) = z n + j=1 (−1)j aj (x)z n−j , where the coefficients aj : Rq , 0 → C (for 1 ≤ j ≤ n) are germs at 0 of smooth functions, S there exists a finite collection T of transformations Ψ : Rq , 0 → Rq , 0, such that {im(Ψ) : Ψ ∈ T } is a neighborhood of 0, which desingularizes the roots of P , i.e., for each Ψ ∈ T , the roots of P ◦ Ψ allow smooth parameterizations near 0; see theorem 3.5. Each Ψ ∈ T is a finite composition of linear coordinate changes and of mappings Rq , 0 → Rq , 0 of the types ϕi and ± ψiN , N ∈ N, where ϕi (x1 , . . . , xq ) = (xi x1 , . . . , xi xi−1 , xi , xi xi+1 , . . . , xi xq ) ± ψiN (x1 , . . . , xq ) = (x1 , . . . , xi−1, ±xN i , xi+1 , . . . , xq ). Note that ϕi is the restriction of a blow-up with center {0} to a coordinate chart and the complexification of + ψiN is an N -sheeted finite branched covering, see 2.7 and 3.1. In order to prove this theorem we present an algorithm which explicitly constructs T . The genericity condition for P is normal nonflatness at 0 (see 2.5) which, roughly speaking, means that the family P (x) does not meet lower dimensional strata of the space of polynomials of degree n with infinite order of flatness at x = 0. Here we use that multiplicities of roots provide a natural stratification SMOOTH MULTIPARAMETER PERTURBATION 3 of the space of polynomials. Essentially the same proof then also provides the following holomorphic version: If the coefficients aj : Cq , 0 → C (for 1 ≤ j ≤ n) are germs at 0 of holomorphic functions, then S there exists a finite collection T of transformations Ψ : Cq , 0 → Cq , 0, such that {im(Ψ) : Ψ ∈ T } is a neighborhood of 0, and, for each Ψ ∈ T , the roots of P ◦ Ψ allow holomorphic parameterizations near 0. Here each Ψ ∈ T is a finite composition of linear coordinate changes and of (complexifications of) mappings of type ϕi and + ψiN . If P is hyperbolic, then linear coordinate changes and transformations of type ϕi suffice for the construction of the Ψ ∈ T . This observation leads to the following result proven in section 4: If P (x) is a smooth family of hyperbolic polynomials which is normally nonflat for each x in a connected smooth manifold M , then there exists a smooth manifold M ′ and a surjective smooth projection Φ : M ′ → M which is a locally finite composition of blow-ups centered at single points such that P ◦ Φ allows locally smooth roots on M ′ . By lifting to the universal covering π : M̃ ′ → M ′ , we obtain global smooth roots of P ◦ Φ ◦ π on M̃ ′ . Evidently, this generalizes the desingularization result in [17]. Theorem 3.5 enables us to prove in section 5 the following generalization of [23, 4.4] mentioned above: If a family of polynomials P has coefficients aj ∈ C ∞(U, C) (for 1 ≤ j ≤ n), where U ⊆ Rq is open, and if P is normally nonflat at x0 ∈ U , then P admits, near x0 , a parameterization λj (for 1 ≤ j ≤ n) of its roots such that the gradient ∇λj ∈ L1 for all j. Simple examples (see 5.4(1)) show that the conclusion in that statement is best possible: In general we cannot expect that the roots of P allow arrangements with first partial derivatives in Lp for any 1 < p ≤ ∞. Compare this theorem with the results obtained in [7] and [8]: For a non-negative real valued function f on an open subset of Rq of class C k (where k ≥ 2) ∇(f 1/k ) belongs to k/(k−2) . L1loc , actually even to Lw The method developed in this paper allows us to give a simple proof of Bronshtein’s result (for slightly stronger conditions): We show that any continuous choice of the roots of a hyperbolic (not necessarily normally nonflat) family P with C n(n+1)/2 coefficients is locally Lipschitz. We think that it is a valuable supplement to the proofs in [5] and [27], which both are rather involved, and to [17], which needs real analytic coefficients. See section 6. In section 7 we give applications to the perturbation theory of normal (resp. Hermitian) matrices and unbounded normal (resp. selfadjoint) operators with compact resolvents and common domain of definition. Our results generalize theorems in perturbation theory of linear operators obtained in [17] and [23]. Consider also [25], [12], [3], [1], [15], [16]. We show the following: For a smooth family U ∋ x 7→ A(x) of normal matrices, where U is a neighborhood of 0 ∈ Rq , there exists a finite collection T of transformations Ψ, with the properties described above, such that, for each Ψ ∈ T , the family A ◦ Ψ allows, locally near 0, smooth parameterizations of its eigenvalues and eigenvectors, provided that A is normally nonflat at 0 in the obvious sense. We can also conclude that the eigenvalues and eigenvectors of normally nonflat A admit, locally, arrangements with first partial derivatives in L1 . An analogous result holds for families of unbounded normal operators in Hilbert space with compact resolvents and common domain of definition. If A(x) is Hermitian and defined and normally nonflat at any x in a connected smooth manifold M , then there exists a smooth manifold M ′ and a surjective smooth projection Φ : M ′ → M which is a locally finite composition of blow-ups centered at single points such that A◦Φ allows smooth eigenvalues and eigenvectors on M ′ , locally. By lifting to the universal covering π : M̃ ′ → M ′ , we obtain global smooth eigenvalues of A ◦ Φ ◦ π on M̃ ′ . For selfadjoint unbounded operators A(x) we obtain a corresponding local result. 4 M. LOSIK, P.W. MICHOR, A. RAINER 2. Preliminaries 2.1. Let P (z) = z n + n X (−1)j aj z n−j = n Y (z − λj ) j=1 j=1 be a monic polynomial with coefficients a1 , . . . , an ∈ C and roots λ1 , . . . , λn ∈ C. By Vieta’s formulas, ai = σi (λ1 , . . . , λn ), where σ1 , . . . , σn are the elementary symmetric functions in n variables: X (2.1) σi (λ1 , . . . , λn ) = λj1 · · · λji . 1≤j1<···<ji ≤n Denote by si , i ∈ N0 , the Newton polynomials elementary symmetric functions by Pn i j=1 λj which are related to the sk − sk−1 σ1 + sk−2 σ2 − · · · + (−1)k−1 s1 σk−1 + (−1)k kσk = 0, (k ≥ 1). (2.2) Let us consider the so-called Bezoutiant   s0 s1 . . . sn−1  s1 s2 . . . sn    B :=  . ..  = (si+j−2 )1≤i,j≤n . . . .. ..  .. .  sn−1 sn . . . s2n−2 Since the entries of B are symmetric polynomials in λ1 , . . . , λn , we find a unique symmetric n × n matrix B̃ with B = B̃ ◦ σ, where σ = (σ1 , . . . , σn). Let Bk denote the minor formed by the first k rows and columns of B. Then we find X (λi1 −λi2 )2 · · · (λi1 −λik )2 · · · (λik−1 −λik )2 . (2.3) ∆k (λ) := det Bk (λ) = i1 <i2 <···<ik ˜ k ◦ σ for unique polySince the polynomials ∆k are symmetric, we have ∆k = ∆ ˜ nomials ∆k . From (2.3) follows that the number of distinct roots of P equals the ˜ k (P ) 6= 0. maximal k such that ∆ Theorem. (Sylvester’s version of Sturm’s theorem, e.g. [22]) Suppose that all coefficients of P are real. Then all roots of P are real if and only if the symmetric n × n matrix B̃(P ) is positive semidefinite. The rank of B̃(P ) equals the number of distinct roots of P and its signature equals the number of distinct real roots. We call a polynomial P with all roots real hyperbolic. Note that roots of a P all hyperbolic polynomial P with a1 = a2 = 0 are equal to 0, since λ2i = s2 (λ) = σ1 (λ)2 − 2σ2 (λ) = a21 − 2a2 . Replacing the variable z by z − a1 /n transforms any polynomial to another one with a1 = 0. Pn 2.2. Splitting lemma. [1, 3.4] Let P0 = z n + j=1 (−1)j aj z n−j be a polynomial satisfying P0 = P1 · P2 , where P1 and P2 are polynomials without common root. Then for P near P0 we have P = P1 (P ) · P2 (P ) for real analytic mappings of monic polynomials P 7→ P1 (P ) and P 7→ P2 (P ), defined for P near P0 , with the given initial values. 2.3. Multiplicity. For a continuous real or complex valued function f defined near 0 in R let the multiplicity m(f) at 0 be the supremum of all integers p such that f(x) = xp g(x) near 0 ∈ R for a continuous function g. Similarly one can define the multiplicity of a function at any x ∈ R. Note that, if f is of class C n and m(f) < n, then f(x) = xm(f) g(x) near 0, where now g is C n−m(f) and g(0) 6= 0. SMOOTH MULTIPARAMETER PERTURBATION 5 We shall say that a continuous real or complex valued function f defined near 0 in Rq is infinitely flat at 0 if inf m(f ◦ ιv ) = ∞, v∈S q−1 where ιv : R → Rq , x 7→ xv. If f is smooth, then this condition is equivalent to the condition that f and its partial derivatives of every order vanish at 0. To see this suppose that f vanishes exactly of finite order m − 1 at 0. That means that the (m − 1)-jet j0m−1 f of f at 0 vanishes, but the m-jet j0m f does not vanish. So j0m f is a homogeneous polynomial of order m on Rq and the set of directions Y ∈ RP q−1 for which j0m f vanishes forms a hypersurface or is empty. Choose v ∈ S q−1 such that the direction R.v does not lie in that hypersurface. Then m(f ◦ ιv ) = m, by Taylor’s formula. The converse is obvious. Actually, for a smooth function f and all its partial derivatives to vanish at 0, it suffices that m(f ◦ ιvi ) = ∞ for a basis vi of Rq . Lemma. [1, 3.7], [23, 2.2] Consider a curve of polynomials P (x)(z) = z n + n X (−1)j aj (x)z n−j , j=2 with aj : R, 0 → C (for 2 ≤ j ≤ n) germs at 0 of smooth functions. Then, for integers r, the following conditions are equivalent: (1) m(ak ) ≥ kr, for all 2 ≤ k ≤ n. ˜ k ◦ P ) ≥ k(k − 1)r, for all 2 ≤ k ≤ n. (2) m(∆ If P is hyperbolic, then (1) and (2) are also equivalent to: (3) m(a2 ) ≥ 2r. Corollary. If the coefficients aj : Rq , 0 → C (for 2 ≤ j ≤ n) are germs at 0 of smooth functions defined in Rq , then the following conditions are equivalent: (1) ak is infinitely flat, for all 2 ≤ k ≤ n. ˜ k ◦ P is infinitely flat, for all 2 ≤ k ≤ n. (2) ∆ If P is hyperbolic, then (1) and (2) are also equivalent to: (3) a2 is infinitely flat. Proof. Apply the above lemma to P ◦ ιv for all v ∈ S q−1 .  Pn 2.4. Stratification. The space Poln of polynomials P (z) = z n + j=1 (−1)j aj z n−j of fixed degree n naturally identifies with Cn , by mapping P to (a1 , . . . , an ). Moreover, Poln may be viewed as the orbit space Cn / Sn of the standard action of the symmetric group Sn on Cn by permuting the coordinates (the roots of P ). In this picture the mapping σ : Cn → Cn , introduced in 2.1, identifies with the orbit projection Cn → Cn / Sn , since the elementary symmetric functions σi in (2.1) generate the algebra C[Cn ]Sn of symmetric polynomials on Cn . The Sn -module Cn and its orbit space Cn / Sn = Poln carry a natural Luna stratification which can described in the following way (e.g. [21]): Let u = (u1 , . . . , un ) ∈ Cn and consider the partition γ(u) of I = {1, 2, . . ., n} into a union of pairwise disjoint nonempty subsets J = J1 ∪ · · · ∪ Js where the numbers i, j ∈ J lie in the same subset precisely when ui = uj . We denote by |γ(u)| the set of integers |J1 |, . . . , |Js | in decreasing order. This set is a partition of the number n. The strata of the Sn module Cn are given by the collection of subsets Cnλ := {u ∈ Cn : |γ(u)| = λ} where λ is a partition of n. The images σ(Cnλ ) constitute the strata of Cn / Sn = Poln . Let s ∈ N0 . Denote by As the union of all strata S of Cn / Sn = Poln with dim S ≤ s, and by Is the ideal of C[Cn / Sn ] ∼ = C[Cn ]Sn generated by 6 M. LOSIK, P.W. MICHOR, A. RAINER ˜ s, ∆ ˜ s+1 , . . . , ∆ ˜ n . Since ∆ σ −1 (As−1 ) = {u ∈ Cn : u has ≤ s − 1 distinct coordinates} = {u ∈ Cn : ∆s (u) = ∆s+1 (u) = · · · = ∆n (u) = 0}, we find that As−1 is the affine√variety associated with Is . Let I˜s denote the radical Is of the ideal Is . We write Rn for the operator of averaging over Sn (the Reynolds operator). For 1 ≤ i1 < · · · < is ≤ n, write Ri1,...,is for the Reynolds operator with respect to the group Si1 ,...,is of permutations of ui1 , . . . , uis . Proposition. The ideal I˜s consists of all sums of Sn -invariant polynomials on Cn of the type Rn (Hfi21 ,...,is ), where H is an Si1 ,...,is -invariant polynomial on Cn and fi1 ,...,is = (ui1 − ui2 ) · · · (ui1 − uis ) · · · (uis−1 − uis ). Proof. It is easy to see that the ideal of polynomials vanishing on σ −1 (As−1 ) is generated by the fi1 ,...,is , for i1 < · · · < is . Evidently, each sum of polynomials of the indicated type lies in I˜s . Let f ∈ I˜s . Then F = f ◦ σ is a Sn -invariant polynomial on Cn which vanishes on σ −1 (As−1 ). Later we identify f with the polynomial F = f ◦ σ. By assumption, we have X F = Gi1,...,is fi1 ,...,is , i1 <···<is with Gi1,...,is ∈ C[Cn ]. Therefore, F = Rn (F ) = X Rn (Gi1 ,...,is fi1 ,...,is ), i1 <···<is and, hence, we can assume that F = Rn (Gfi1 ,...,is ), where G ∈ C[Cn ]. For τ ∈ Si1 ,...,is we have τ.fi1 ,...,is = sgn(τ )fi1 ,...,is . It follows that Rn (Gfi1 ,...,is ) = Rn (Ri1,...,is (Gfi1 ,...,is )) = Rn (Alt(i1 , . . . , is )(G)fi1 ,...,is ), where Alt(i1 , . . . , is ) is the alternation operator with respect to the group Si1 ,...,is . This implies that F = Rn (Hfi21 ,...,is ),  where H is a Si1 ,...,is -invariant polynomial on Cn . It follows that Is does not coincide with its radical I˜s : e.g. the polynomial 2 Rn ((x21 + · · · + x2s )f1,...,s ) is not contained in Is . 2.5. Normal nonflatness. Let U ⊆ Rq be an open neighborhood of 0. We call a family of monic polynomials n X (−1)j aj (x)z n−j , P (x)(z) = z n + j=1 k with smooth (or just C for some k ≥ 0) coefficients aj : U → C (for 1 ≤ j ≤ n) normally nonflat at 0 if one of the following two equivalent conditions is satisfied: (1) Let s be a minimal integer such that, for a neighborhood Ω ⊆ U of 0 ∈ Rq , ˜ s (P (x)) is not infinitely flat at 0. we have P (Ω) ⊆ As . Then x 7→ ∆ ˜ s (P (x)) (2) Let s be maximal with the property that the germ at 0 of x 7→ ∆ ˜ s (P (x)) is not infinitely flat at 0. is not 0. Then x 7→ ∆ The equivalence is immediate, since the integers s introduced in (1) and (2), respectively, coincide due to the fact that As is the affine variety associated with Is+1 . Note that these conditions are automatically satisfied if the coefficients aj are analytic or, more generally, quasianalytic. SMOOTH MULTIPARAMETER PERTURBATION 7 Similarly, normal nonflatness is defined at any x ∈ U . Since it is invariant under diffeomorphisms of U fixing x, normal nonflatness is also well defined for polynomials P whose smooth coefficients are defined on a smooth manifold. If q = 1, continuous parameterizations of the roots of P exist (e.g. [12, II 5.2]). In that case (1) and (2) are also equivalent to the following condition: (3) If two of the continuously arranged roots of P meet of infinite order of flatness at 0, then their germs at 0 are equal. This is a consequence of (2.3) and the following claim: Let f, g be real or complex valued continuous functions defined near 0 ∈ R and not infinitely flat at 0. Then the product f · g is not infinitely flat at 0. For contradiction assume that m(f · g) = ∞ while (without loss) m(f) = m(g) = 0. For each integer p ≥ 0 there is a continuous function hp such that f(x)g(x) = xphp (x) near 0. Now f(x)/x2 and, analogously, g(x)/x2 are unbounded as x → 0, since otherwise f(x) = xf˜(x) with continuous f˜(x) := xf(x)/x2 , x 6= 0, and f˜(0) := 0, which contradicts m(f) = 0. But then in f(x)g(x)/x4 = h4 (x) the left-hand side is unbounded while the right-hand side is bounded as x → 0, a contradiction. We compare this notion of normal nonflatness with the notion introduced in [2] (actually for curves): (1′ ) Let s be a minimal integer such that, for a neighborhood Ω ⊆ U of 0 ∈ Rq , we have P (Ω) ⊆ As . Then, for some f ∈ I˜s , the function x 7→ f(P (x)) is not infinitely flat at 0. Proposition. (1) and (1′ ) are equivalent. Proof. We prove the non-trivial direction (1′ ) ⇒ (1). By (1′ ), for some v ∈ S q−1 the function f ◦ P ◦ ιv has finite multiplicity. Let λj be a continuous choice of the roots of P ◦ ιv . By the minimality of s, there are exactly s pairwise distinct roots among the λj near 0. By proposition 2.4 we have ˜ s ) ◦ P ◦ ιv f ◦ P ◦ ιv = (H(λj ) · ∆ ˜ s ◦ P ◦ ιv has finite multiplicity, for some polynomial H on Cn . This implies that ∆ ˜ and so ∆s ◦ P is not infinitely flat at 0.  As a consequence the following lemma can be proved in analogy with [2, 3.5]. We give also an independent proof. Lemma 2.6. Application of the splitting lemma preserves normal nonflatness. Proof. Let P be normally nonflat at 0. Suppose that the roots of P (0) split into two nonempty disjoint subsets of cardinalities n1 and n2 , n1 + n2 = n, which defines a splitting P (x) = P1 (x)P2 (x) for x near 0, according to lemma 2.2. Let s1 be ˜ s (P1 ) is not 0. Assume for maximal with the property that the germ at 0 of ∆ 1 ˜ contradiction that ∆s1 (P1 ) is infinitely flat at 0. By 2.3, this is the case if and only ˜ s1 (P1 ◦ ιv ) is infinitely flat at 0 for each v ∈ S q−1 . Moreover, there is a v ∈ S q−1 if ∆ ˜ s1 (P1 ◦ ιv ) is not 0. Let such a v be fixed. Choose such that the germ at 0 of ∆ a continuous parameterization of the roots of P1 ◦ ιv . It follows that at least two roots of P1 ◦ ιv with distinct germs at 0, and hence also of P ◦ ιv , meet of infinite order of flatness at 0. Let s be maximal with the property that the germ at 0 of ˜ s (P ) is not 0. Then ∆ ˜ s (P ◦ ιv ) vanishes of infinite order of flatness at 0. For ∆ ˜ s (P ◦ ιv ) ˜ those v, where ∆s1 (P1 ◦ ιv ) vanishes identically near 0, obviously also ∆ ˜ vanishes identically near 0. So ∆s (P ) vanishes of infinite order of flatness at 0, a contradiction.  8 M. LOSIK, P.W. MICHOR, A. RAINER 2.7. Blow-up. Let K stand for R or C. Let V be an open neighborhood of 0 ∈ Kq and put V ′ := {(x, l) ∈ V × KP q−1 : x ∈ l}. The mapping ϕ : V ′ → V defined by ϕ(x, l) = x is called blow-up of V with center {0}. Then ϕ is proper, restricts to a homeomorphism over V \{0}, and ϕ−1 (0) = KP q−1 . Let x = (x1 , . . . , xq ) denote affine coordinates of Kq , and let ξ = [ξ1 , . . . , ξq ] denote homogeneous coordinates of KP q−1 . Then V ′ = {(x, ξ) : xi ξj = xj ξi , 1 ≤ i, j ≤ q} can be covered by coordinate charts Vi′ := {(x, ξ) ∈ V ′ : ξi 6= 0}, for 1 ≤ i ≤ q, with coordinates (xi1 , . . . , xiq ) where xii = xi and xij = ξj /ξi , j 6= i. With respect to these coordinates, ϕ is given by (xi1 , . . . , xiq ) 7→ (xii xi1 , . . . , xiixii−1 , xii , xii xii+1 , . . . , xii xiq ). We shall also consider blow-ups of open neighborhoods U of points x in a smooth (or analytic) manifold M with center {x}. Using local coordinates on M , it can evidently be referred back to blow-ups of open neighborhoods of 0 ∈ Kq with center {0}. 3. Smooth perturbation of polynomials 3.1. The transformations ϕi and ψiN . Let ϕi : Rq , 0 → Rq , 0 denote the transformation ϕi (x1 , . . . , xq ) = (xi x1 , . . . , xi xi−1 , xi , xi xi+1 , . . . , xi xq ), and let ± ψiN : Rq , 0 → Rq , 0 be given by ± ψiN (x1 , . . . , xq ) = (x1 , . . . , xi−1 , ±xN i , xi+1 , . . . , xq ). It is clear that the complexification of + ψiN is an N -sheeted finite branched covering of Cq with branch point 0 ∈ Cq . If N is even, + ψiN is not surjective. In that case we try to remedy this fact by using + ψiN and − ψiN for xi ≥ 0 and xi ≤ 0, respectively. When N is odd, it suffices to use just + ψiN . The mapping ϕi identifies with the restriction of a blow-up of an open neighbor′ hood V of 0 ∈ Rq with center {0} to the distinct coordinate 2.7. Pn chartj Vi , seen−j n , with Consider a family of polynomials P (x)(z) = z + j=1 (−1) aj (x)z aj : Rq , 0 → C (for 1 ≤ j ≤ n) germs at 0 of smooth functions and normally nonflat at 0. We claim that normal nonflatness is preserved under pullback by transformations ϕi and ± ψiN , i.e., the families of polynomials P ◦ ϕi and P ◦ (±ψiN ) are normally nonflat at 0 as well. Let s be minimal with the property that the germ ˜ s ◦ P is not 0. Let U ⊆ Rq be a neighborhood of 0. Evidently, there exist at 0 of ∆ q−1 vi ∈ S which form a basis of Rq and ǫ > 0 such that the images ϕi (U ) and ± N ψi (U ) contain the segments [0, ǫ].vi. Since P is normally nonflat at 0, the germ ˜ s ◦P ◦ϕi and of ∆ ˜ s ◦P ◦(± ψN ) is not 0, and s is maximal with that property. at 0 of ∆ i ˜ s ◦ P ◦ ϕi nor It is easy to verify (e.g. with Faà di Bruno’s formula) that neither ∆ ± N ˜ ∆s ◦ P ◦ ( ψi ) are infinitely flat at 0. Lemma 3.2. Let f1 , . . . , fp : Rq , 0 → C be germs at 0 of smooth (or C k with sufficiently large k) functions such that fj vanishes exactly of order mj − 1 at 0 and let fp+1 , . . . , fn : Rq , 0 → C be infinitely flat smooth (or C k ) germs at 0. Then there exists a (Zariski) open subset in GL(q, R) such that for each A in this set we have: SMOOTH MULTIPARAMETER PERTURBATION 9 (1) smooth (or C k−mj ) germs gj,i : Rq , 0 → C with gj,i (0) 6= 0 such that m (fj ◦ A ◦ ϕi )(x) = xi j .gj,i(x), x = (x1 , . . . , xq ) for 1 ≤ j ≤ p and 1 ≤ i ≤ q; (2) smooth (or C k−mj ) (flat) germs gj,i : Rq , 0 → C such that m (fj ◦ A ◦ ϕi )(x) = xi j gj,i (x) for p + 1 ≤ j ≤ n, 1 ≤ i ≤ q, and each 0 ≤ mj (≤ k). Proof. Suppose that all fj (for 1 ≤ j ≤ n) are defined on an open convex neighm −1 borhood of 0. By assumption, the (mj − 1)-jet j0 j fj of fj at 0 vanishes, but the mj m mj -jet j0 fj does not vanish, for 1 ≤ j ≤ p. So j0 j fj is a homogeneous polynomial m of order mj on Rq . The directions Y ∈ RP q−1 for which j0 j fj vanishes, form a mj real algebraic variety V (j0 fj ) ⊂ RP q−1 , a hypersurface of degree mj . It might be empty. Now we choose q linearly independent unit vectors y1 , . . . , yq ∈ S q−1 such that S m the directions R.yi do not lie in 1≤j≤p V (j0 j fj ). Choose a linear coordinate change A for which y1 , . . . , yq is the new basis of Rq . We show the statement for i = 1; the other cases are analogous. Let us use the notation x = (x1 , . . . , xq ) = mj −1 (t, z) where t = x1 and z = (x2 , . . . xq ). Since ϕ1 (0, z) = (0, 0) we have j(0,z) (fj ◦ m −1 m −1 j (fj ◦ A) • j0,zj ϕ1 = 0 where • denotes truncated composition. So A ◦ ϕ1 ) = j(0,0) fj ◦ A ◦ ϕ1 (for 1 ≤ j ≤ p) vanishes of order mj − 1 at each (0, z) and we can use the Taylor integral remainder of order mj of t 7→ (fj ◦ A ◦ ϕ1 )(t, z) as follows: Z 1 (1 − u)mj −1 ∂ mj (fj ◦ A ◦ ϕ1 ) (ut, z) du (fj ◦ A ◦ ϕ1 )(t, z) = tmj (mj − 1)! ∂tmj 0 gj,1 (0, 0) 6= 0. =: tmj .gj,1 (t, z), The statement for the flat germs fj (for p + 1 ≤ j ≤ n) is immediate.  Lemma 3.3. Let ak : Rq , 0 → C (for 2 ≤ k ≤ n) be germs at 0 of smooth (or C l ) k functions. Suppose that ak (x) = xm i bk,i (x), for 2 ≤ k ≤ n and for fixed i, positive p integers mk , and smooth (or C ) germs bk,i : Rq , 0 → C such that bk,i (0) 6= 0, for some k. Let nm o k m := min : bk,i (0) 6= 0 . k and assume that, for those k with bk,i(0) = 0, we have mk − km ≥ 0 as well. Then there exist positive integers N and M such that ± ψN i ak (± ψiN (x)) = xkM i ak,(M ) (x), ± ψN 2 ≤ k ≤ n, ± ψN i q i for smooth (or C p ) germs ak,(M ) : R , 0 → C with ak,(M ) (0) 6= 0 for some k. If the Pnak (for 2 ≤ k ≤ n) are the coefficients of a hyperbolic polynomial P (x)(z) = z n + j=2 (−1)j aj (x)z n−j , then in the former statement N = 1, more precisely, there exists a positive integer M such that ak (x) = xkM i ak,i,(M ) (x), 2 ≤ k ≤ n, for smooth (or C p ) germs ak,i,(M ) : Rq , 0 → C with ak,i,(M )(0) 6= 0 for some k, in particular a2,i,(M )(0) 6= 0. Proof. Let d be a minimal integer such that dm ≥ 1. Then dmk ≥ dmk ≥ k and hence we may write, for all k, ± ψd k i ak (± ψid (x)) = (±1)mk xdm bk,i (± ψid (x)) = xki ak,(d) (x) i ± ψd i with smooth (or C p ) germs ak,(d) : Rq , 0 → C, for 2 ≤ k ≤ n. 10 M. LOSIK, P.W. MICHOR, A. RAINER ± ψd i We have ak,(d) (x) = (±1)mk xidmk −k bk,i(± ψid (x)), for 2 ≤ k ≤ n. Note that bk,i (0) 6= 0 if and only if bk,i (± ψid (0)) 6= 0. Thus   dmk − k ± d : bk,i ( ψi (0)) 6= 0 = dm − 1 < m, (3.1) m(d) := min k by the minimality of d. ± d ψi (0) 6= 0, and we are done. Otherwise If m(d) = 0 there is some k such that ak,(d) ± ψd i , for 2 ≤ k ≤ n, in the place of the ak . we repeat the procedure with ak,(d) Since m(d) is of the form p/k where 2 ≤ k ≤ n and p ∈ N0 and by (3.1), this algorithm stops after finitely many steps, which proves the first statement. If ak for 2 ≤ k ≤ n are the coefficients of a hyperbolic polynomial P , we may ˜ 2 ◦ P = −2na2 , by theorem 2.1, assume b2,i (0) 6= 0, by corollary 2.3. Since 0 ≤ ∆ we find m2 = 2M for a positive integer M . For contradiction assume that m < M . Consider the following continuous family of polynomials for xi ≥ 0: n n X X m −jm j −jm n−j n n P(m) (x)(z) := z + (−1)j xi j bj,i (x)z n−j . (−1) xi aj (x)z =z + j=2 j=2 x−m λj (x) i If λj (x) are the roots of P (x), then are the roots of P(m) (x), for xi > 0. So, for xi > 0, P(m) (x) is hyperbolic as well. Since the space of hyperbolic polynomials is closed (e.g. theorem 2.1), also P(m) (0) is hyperbolic. By m2 − 2m = 2M − 2m > 0, all roots of P(m) (0) are equal to 0. This is a contradiction for those k with mk = km and bk,i (0) 6= 0.  3.4. Reduction to smaller permutation groups. In the proof of theorem 3.5 below we shall reduce our perturbation problem in virtue of the splitting lemma 2.2. Let U ⊆ Rq be an Pnopen neighborhood of 0 and consider a family of polynomials P (x)(z) = z n + j=1 (−1)j aj (x)z n−j , with smooth coefficients aj : U → C for 1 ≤ j ≤ n, which is normally nonflat at 0. If there are distinct roots ν1 , . . . , νl of P (0), the splitting lemma 2.2 provides a factorization P (x) = P1 (x) · · · Pl (x) near 0 such that all roots of Ph (0) equal νh , for 1 ≤ h ≤ l. According to the interpretation in 2.4, this factorization amounts to a reduction of the Sn -action on Cn to the Sn1 × · · · × Snl -action on Cn1 ⊕ · · · ⊕ Cnl , where nh is the multiplicity of νh for 1 ≤ h ≤ l. By lemma 2.6, normal nonflatness is preserved by this reduction. nl n1 Further, we may remove fixed points of the Sn1 × · · ·×SP nl -action on C ⊕· · ·⊕C n nh j nh −j or, equivalently, reduce each factor Ph (x)(z) = z + j=1 (−1) ah,j (x)z to ˜ the case ah,1 = 0 by replacing z by z − ah,1 (x)/nh . In view of (2.3) the ∆k (Ph ) are not affected by this change of variables, and hence normal nonflatness is preserved. If P is hyperbolic, the Sn -module Rn is used instead of Cn . N ,...,N p 3.5. Let us fix notation: We write Ψ±i11 ,...,±i , where 1 ≤ i1 , . . . , ip ≤ q and Nj ∈ N, p q q for the transformation R , 0 → R , 0 composed by p pieces A ◦ ϕi ◦ (± ψiN ), where sub- and superscript indicate the appearance of the single pieces with sign, degree N , and in its order. Here A is a linear coordinate change and ϕi and ± ψiN are as in 3.1. Theorem. Consider a family of polynomials n X (−1)j aj (x)z n−j , P (x)(z) = z n + j=1 q with aj : R , 0 → C (for 1 ≤ j ≤ n) germs at 0 of smooth functions, which is N ,...,Np normally nonflat at 0. There exists a finite collection T = {Ψ±i11 ,...,±i : 1 ≤ p SMOOTH MULTIPARAMETER PERTURBATION 11 S i1 , . . . , ip ≤ q} such that {im(Ψ) : Ψ ∈ T } is a neighborhood of 0 and for each Ψ ∈ T the family of polynomials P ◦ Ψ allows smooth roots near 0, i.e., there exist smooth germs λ1 , . . . , λn : Rq , 0 → C such that P (Ψ(x))(z) = z n + n X (−1)j aj (Ψ(x))z n−j = n Y j=1 j=1 (z − λj (x)). If P is hyperbolic, then for the composition of each Ψ ∈ T suffice linear coordinate changes and transformations of type ϕi . Proof. We use the following algorithm: (1) If all roots of P (0) are pairwise different, the roots of x 7→ P (x) may be parameterized smoothly near 0, by the implicit function theorem. (2) If there are multiple distinct roots ν1 , . . . , νl of P (0) (we allow also l = 1), the splitting lemma 2.2 provides a factorization P (x) = P1 (x) · · · Pl (x) near 0 such that all roots of Ph (0) equal νh , for 1 ≤ h ≤ l. As indicated in 3.4, we reduce to the Sn1 × · · · × Snl -action on Cn1 ⊕ · · · ⊕ Cnl , where nh is the multiplicity of νh for 1 ≤ h ≤ l, and we remove fixed points. (3) All roots of Ph (0), for 1 ≤ h ≤ l, are equal to 0, hence, ah,k (0) = 0 for all 1 ≤ k ≤ nh . If all coefficients ah,k , for 1 ≤ k ≤ nh , of Ph are identically 0, we choose λh,k = 0 for all 1 ≤ k ≤ nh . Without loss assume that for all 1 ≤ h ≤ l there are 1 ≤ k ≤ nh such that ah,k 6= 0. By corollary 2.3, for each h not all ah,k , for 1 ≤ k ≤ nh , are infinitely flat at 0. By lemma 3.2, there exists a linear coordinate change A on Rq , 0, positive integers mh,k , and smooth germs bh,k,i , for 1 ≤ h ≤ l, 2 ≤ k ≤ nh , and 1 ≤ i ≤ q, such that mh,k (ah,k ◦ A ◦ ϕi )(x) = xi bh,k,i (x), (3.2) where bh,k,i (0) 6= 0 for exactly those h and k for which ah,k is not infinitely flat at 0. We put nm o h,k m := min : bh,k,i (0) 6= 0 . k We may suppose that, for those h and k with ah,k infinitely flat, mh,k is chosen such that mh,k − km ≥ 0. By lemma 3.3, there exist integers N and M such that A◦ϕ ◦(± ψiN ) i (ah,k ◦ A ◦ ϕi ◦ (± ψiN ))(x) = xkM i ah,k,(M ) (x), 1 ≤ h ≤ l, 2 ≤ k ≤ nh , 1 ≤ i ≤ q, A◦ϕ ◦(± ψ N ) A◦ϕ ◦(± ψ N ) i i i i for smooth germs ah,k,(M : Rq , 0 → C with ah,k,(M (0) 6= 0 for each h and ) ) some 2 ≤ k ≤ nh . Let us consider, for 1 ≤ h ≤ l and 1 ≤ i ≤ q, the following family of polynomials A◦ϕ ◦(± ψiN ) Ph,(Mi) (x)(z) := z nh + nh X A◦ϕ ◦(± ψiN ) i (−1)j ah,j,(M ) (x)z nh −j j=2 which is normally nonflat at 0, since ˜ k (P A◦ϕi ◦( ˜ k ((Ph ◦ A◦ ϕi ◦ (± ψN ))(x)) = xk(k−1)M ∆ ∆ i i h,(M ) ± ψiN ) (x)), 2 ≤ k ≤ nh . (3.3) Moreover, we put A◦ϕ ◦(± ψiN ) P(M ) i (x) := l Y A◦ϕ ◦(± ψiN ) Ph,(Mi) (x). (3.4) h=1 A◦ϕ ◦(± ψ N ) i (x) is smoothly solvable and x 7→ λh,j (x) are its smooth roots, If x 7→ Ph,(Mi) then x 7→ xM λ (x) form a choice of smooth roots of x 7→ (Ph ◦ A ◦ ϕi ◦ (± ψiN ))(x). h,j i A◦ϕ ◦(± ψiN ) By construction, not all roots of Ph,(Mi) (0) coincide. So we may feed 12 M. LOSIK, P.W. MICHOR, A. RAINER A◦ϕ ◦(± ψ N ) i , for 1 ≤ i ≤ q, into the algorithm, reminding the splitting (3.4), P(M ) i i.e., we perform the algorithm for the single factors separately but simultaneously. This algorithm stops after finitely many steps and yields the assertion. In order to show the supplement in the theorem, it suffices to prove that for hyperbolic P we do not need transformations of type ± ψiN in step (3) of the above algorithm. Suppose that we have a splitting P (x) = P1 (x) · · · Pl (x) near 0 (where l = 1 is allowed) and that all roots of Ph (0), for 1 ≤ h ≤ l, are equal to 0, hence, ah,k (0) = 0 for all 1 ≤ k ≤ nh . Assume that for each h there exists a 1 ≤ k ≤ nh such that ah,k 6= 0. Since Ph is hyperbolic, ah,2 6= 0, and, by corollary 2.3, ah,2 is not infinitely flat at 0. So, by lemma 3.2, there exists a linear coordinate change A on Rq , 0, positive integers mh,k , and smooth germs bh,k,i , for 1 ≤ h ≤ l, 2 ≤ k ≤ nh , and 1 ≤ i ≤ q, such that (3.2) holds, where in particular bh,2,i (0) 6= 0 for all h and ˜ 2 ◦ Ph = −2nah,2 , by theorem 2.1, we find mh,2 = 2Mh for a positive i. Since 0 ≤ ∆ integer Mh . Put M := min{Mh : 1 ≤ h ≤ l}. By lemma 3.3, A◦ϕi (ah,k ◦ A ◦ ϕi )(x) = xkM i ah,k,(M ) (x), 1 ≤ h ≤ l, 2 ≤ k ≤ nh , 1 ≤ i ≤ q, A◦ϕi i q for smooth germs aA◦ϕ h,k,(M ) : R , 0 → R with ah,k,(M ) (0) 6= 0 for some h, k, and i in particular, for some h, we have aA◦ϕ h,2,(M ) (0) 6= 0. We consider the family of hyperbolic polynomials A◦ϕi Ph,(M ) (x)(z) n := z + n X n−j i (−1)j aA◦ϕ h,j,(M ) (x)z j=2 which is normally nonflat at 0, by (the analog of) (3.3), and we define A◦ϕi (x) := P(M ) l Y A◦ϕi (x). Ph,(M ) (3.5) h=1 A◦ϕi If x 7→ P(M ) (x) is smoothly solvable, then x 7→ (P ◦ A ◦ ϕi )(x) is smoothly solvable A◦ϕi i as well. Since aA◦ϕ h,2,(M ) (0) 6= 0 for some h, not all roots of Ph,(M ) (0) coincide, and A◦ϕi we feed P(M ) , for 1 ≤ i ≤ q, into the algorithm.  Remark 3.6. The transformation ϕi : Rq , 0 → Rq , 0 is not surjective: points x 6= 0 with xi = 0 do not lie in the image of ϕi . Studying the perturbation of polynomials P , we therefore loose information by composing P with a single ϕi . But we are able to correct this loss of information by considering the set of all transformations ϕi (for 1 ≤ i ≤ q) simultaneously, which is guaranteed by lemma 3.2. In the hyperbolic case we can even say more, see theorem 4.2. The transformation + ψiN : Rq , 0 → Rq , 0 is surjective only when N is odd. In order to study perturbations of polynomials we can remedy the situation, for even N , by dealing with the cases xi ≥ 0 and xi ≤ 0, separately, using + ψiN and − ψiN , respectively. The necessary information is encoded in the collection T , provided by theorem 3.5. This will be illustrated in the following simple example. √ Example. We consider P (x1 , x2 )(z) = z 2 −(x1 +ix2 ) with x1 , x2 ∈ R and i = −1. The function f(x1 , x2) := x1 + ix2 vanishes of order 0 at 0, and (in the standard coordinates) we have (f ◦ ϕ1 )(x1 , x2 ) = x1 (1 + ix2 ), (f ◦ ϕ2 )(x1 , x2 ) = x2 (x1 + i). SMOOTH MULTIPARAMETER PERTURBATION 13 Furthermore, (f ◦ ϕ1 ◦ (± ψ12 ))(x1 , x2 ) = (f ◦ Ψ2±1 )(x1 , x2 ) = ±x21 (1 ± ix2 ), (f ◦ ϕ2 ◦ (± ψ22 ))(x1 , x2 ) = (f ◦ Ψ2±2 )(x1 , x2 ) = ±x22 (±x1 + i) allow smooth square roots for x1 , x2 near 0. We choose them in the following way: k=1 k=2 xk > 0 q √ ± f ◦ Ψ2+1 = ±x1 1 + ix2 q √ ± f ◦ Ψ2+2 = ±x2 x1 + i xk < 0 q √ ± f ◦ Ψ2−1 = ±x1 −1 + ix2 q √ ± f ◦ Ψ2−2 = ±x2 x1 − i 3.7. Holomorphic perturbation. Carrying out the obvious modifications in the proof of theorem 3.5 we obtain (where ϕi and + ψiN are replaced by their complexifications): Theorem. Consider a family of polynomials n X (−1)j aj (x)z n−j , P (x)(z) = z n + j=1 q with aj : C , 0 → C (for 1 ≤ j ≤ n) germs at 0 of holomorphic functions. There S N ,...,N exists a finite collection T = {Ψi11,...,ip p : 1 ≤ i1 , . . . , ip ≤ q} such that {im(Ψ) : Ψ ∈ T } is a neighborhood of 0 and for each Ψ ∈ T the family of polynomials P ◦ Ψ allows holomorphic roots near 0.  4. Smooth perturbation of hyperbolic polynomials Proposition 4.1. Let M be a connected smooth manifold. Consider a family of hyperbolic polynomials n X (−1)j aj (x)z n−j , P (x)(z) = z n + j=1 with aj : M → R (for 1 ≤ j ≤ n) smooth functions, which is normally nonflat at any x ∈ M . Let s be a minimal integer such that P (M ) ⊆ As . Suppose that there exists a smooth parameterization λ1 , . . . , λs of the distinct roots of P on M . Then each smooth root of P on M equals one of the λi and any two choices of smooth roots of P on M differ by a permutation. Proof. The subset P −1 (As \As−1 ) of M is evidently open and, by normal nonflat˜ s (P (x)) = 0} is ness, also dense in M : the interior of P −1 (As−1 ) = {x ∈ M : ∆ empty. The restriction of the projection σ (see 2.1) to each connected component of σ −1 (As \As−1 ) is a diffeomorphism onto As \As−1 . It follows that the s distinct roots of P allow a smooth parameterization λi on each connected component of P −1 (As \As−1 ). These differ by permutations. Suppose that M ⊆ R is an open interval. Then the s distinct roots of P allow a smooth parameterization on M which is unique up to permutations. This is due to [1]. For convenience we give a proof. Locally, the existence of a smooth parameterization of the roots follows from theorem 3.5. We show that a smooth choice λi of the roots is unique up to permutations. Hence the local smooth roots may be glued to global smooth roots which are unique up to permutations. P −1 (As \As−1 ) is open and dense in M , i.e., a union of adjoining open intervals. Let x ∈ P −1 (As−1 ) and let (a, x) and (x, b) be the neighboring intervals. Let µ be a smooth root of P defined near x. By the above, the restrictions of µ to (a, x) and (x, b) are equal 14 M. LOSIK, P.W. MICHOR, A. RAINER to the restrictions of λi and λj , respectively. If i 6= j, then λi − λj is flat at x, a contradiction by 2.5(3). Thus, i = j, and µ equals one of the λi . In the general case let x ∈ M with P (x) ∈ As−1 . Suppose that on a neighborhood V of x there exists a smooth parameterization λ1 , . . . , λs of the s distinct roots of P . Then the λi are uniquely determined up to permutations, locally near x. We consider the blow-up ϕ : V ′ → V with center {x} and the family of hyperbolic polynomials P ◦ ϕ on V ′ . By 2.7, we can assume that V is an open neighborhood of x = 0 in Rq , where q = dim M . By lemma 3.2, applied to the coefficients of P ˜ s ◦ P , choosing appropriate linear coordinates in Rq , provides us with local and to ∆ coordinates on V ′ (again denoted by xi ) such that on each chart Vi′ (see 2.7) the ˜ s ◦ P ◦ ϕi have the form coefficients of the restriction P ◦ ϕi = P ◦ ϕ|Vi′ and ∆ k (ak ◦ ϕi )(x) = xm i bk,i (x), ˜ s ◦ P ◦ ϕi )(x) = (∆ for all k, xm i Γi (x), for positive integers mk and m and smooth functions bk,i and Γi , where bk,i (0) 6= 0, for some k, and Γi (0) 6= 0. Shrinking V if necessary, we can assume that those bk,i ˜ s ◦ P ◦ ϕi = 0 is equivalent and Γi do not vanish on Vi′ . Then, on Vi′ , the equation ∆ to the equation xi = 0. We consider a curve c(t) in Vi′ given by the equations xi = t, xj = cj for j 6= i, where the cj are sufficiently small non-zero constants. It follows that the curve of polynomials P ◦ ϕi ◦ c is normally nonflat at 0. Let µ be a smooth root of P on V . To prove that µ equals on of the λk it suffices to prove that µ ◦ ϕ equals one of the λk ◦ ϕ. As seen above, the restrictions of λk ◦ ϕi to the half spaces xi < 0 and xi > 0 coincide with the restrictions of some λj ◦ ϕi and λk ◦ ϕi . By the one-dimensional case applied to P ◦ ϕi ◦ c we find j = k. So µ ◦ ϕi coincides with one of the λk ◦ ϕi on Vi′ . Since this is true for any i, we get the assertion. Finally, we prove the proposition. Let µ be a smooth roots of P on M and let Uβ denote the connected components of P −1 (As \As−1 ) which is open and dense in M . The restriction of µ to each Uβ coincides with the restriction of some λi . The assertion above applied to a point x ∈ U β1 ∩ U β2 implies that µ coincides with one λi on U β1 ∪ U β2 . The statement follows.  Theorem 4.2. Let M be a connected smooth manifold. Consider a family of hyperbolic polynomials n X (−1)j aj (x)z n−j , P (x)(z) = z n + j=1 with aj : M → R (for 1 ≤ j ≤ n) smooth functions, which is normally nonflat at any x ∈ M . Then there exists a smooth manifold M ′ and a surjective smooth projection Φ : M ′ → M which is a locally finite composition of blow-ups centered at single points such that P ◦ Φ locally allows smooth roots on M ′ . If π : M̃ ′ → M ′ is the universal covering of M ′ , then P ◦ Φ ◦ π admits smooth roots on M̃ ′ , globally. Proof. Let s be a minimal integer such that P (M ) ⊆ As . Recall that P −1 (As \As−1 ) is open and dense in M . If x ∈ P −1 (As \As−1 ), we denote by Ux an open neighborhood of x such that Ux ⊆ P −1 (As \As−1 ). If x ∈ P −1 (As−1 ), we use a blow-up with center {x}: We apply lemma 3.2 to the coefficients of P and choose an open neighborhood Ux of x sufficiently small such that for each coordinate chart (see ′ 2.7) Ux,i of the blow-up ϕx : Ux′ → Ux the reduced polynomials (3.5), obtained by the algorithm in 3.5, are defined. The family {Ux } is an open covering of M . Let {Uα } be a locally finite open covering of M subordinate to {Ux }. We may assume that all Uα are convex with respect to some Riemannian metric on M . For those Uα subordinate to a Ux with x ∈ P −1 (As−1 ) we use a blow-up ϕα : Uα′ → Uα , as explained above. Shrinking the Uα if necessary, we can suppose that the center of SMOOTH MULTIPARAMETER PERTURBATION 15 each blow-up ϕα is covered only by Uα and none of the other Uβ . Replacing in M the blown up Uα by Uα′ , provides a smooth manifold M1 and a surjective smooth projection Φ1 : M1 → M . The reduced polynomials (3.5), associated to P ◦ Φ1 , are defined on some locally finite covering of M1 . We apply the same procedure to M1 and the reduced polynomials associated to P ◦ Φ1 which provides a smooth manifold M2 , a surjective smooth projection Φ2 : M2 → M1 , and reduced polynomials associated to P ◦ Φ1 ◦ Φ2 . Proceeding like this yields after a finite number k of such steps a smooth manifold M ′ := Mk and a surjective smooth projection Φ := Φk ◦ · · · ◦ Φ1 : M ′ → M . By construction, there is a locally finite open covering {Vβ } of M ′ such that Φ|Vβ is a finite composition of blow-ups centered at single points, for each β, and P ◦ Φ admits smooth parameterizations of its roots on M ′ , locally. By construction, P ◦ Φ is normally nonflat at any x ∈ M ′ , and there is a locally finite open covering {Wα } of M ′ such that, for each α, P ◦ Φ allows smooth roots on Wα . We may suppose that each Wα is convex with respect to some Riemannian metric on M ′ . Let Wα ∩ Wβ 6= ∅. Suppose (without loss) that Wα ∩ Wβ is connected. By proposition 4.1, smooth choices of the roots of P ◦ Φ|Wα and P ◦ Φ|Wβ , respectively, differ by a permutation on Wα ∩ Wβ . Therefore, we can glue them to smooth roots on Wα ∪ Wβ . By lifting to the universal covering π : M̃ ′ → M ′ , we are able to glue the local smooth roots to global smooth roots of P ◦ Φ ◦ π on M̃ ′ .  Remark. A similar local theorem for real analytic coefficients aj was proved in [17]. The method of proof is different. 5. Roots with first order partial derivatives in L1loc We show in this section that the roots of an everywhere normally nonflat family of polynomials n X (−1)j aj (x)z n−j , P (x)(z) = z n + j=1 ∞ with coefficients aj ∈ C (U, C) (for 1 ≤ j ≤ n), where U ⊆ Rq is open, allow a parameterization λi (for 1 ≤ j ≤ n) such that ∇λi ∈ L1loc for all i. 5.1. The class W. Let U be an open subset of Rq . We denote by W(U ) the class of all real or complex valued functions f with the following properties: (i) f is defined and C ∞ on the complement of a finite union EU of hypersurfaces in U . (ii) f is bounded on U \EU . (iii) The first partial derivatives of f are in L1 (U ). Alternatively to condition (ii) it is possible to use: (ii’) f ∈ L1 (U ). We shall also use W(U, Cn ), where f = (f1 , . . . , fn ) belongs to W(U, Cn ) iff each fi belongs to W(U ). 5.2. Let us write Ω := {x ∈ Rq : |xj | < 1 for all j}, Ωxi >0 := {x ∈ Rq : 0 < xi < 1, |xj | < 1 for j 6= i}, Ωx̂i := {(x1 , . . . , x̂i , . . . , xq ) : x ∈ Ω}, Ω̃i := {x ∈ Rq : 0 < |xi | < 1, |xj | < |xi | for j 6= i}, Ω̃x̂i := {(x1 , . . . , x̂i , . . . , xq ) : x ∈ Ω̃i }. 16 M. LOSIK, P.W. MICHOR, A. RAINER Sq Note that Ω̃i is a simplicial double cone and Ω = i=1 Ω̃i . Recall the transformations ϕi and ± ψiN from 3.1. We have ϕi (Ω\{xi = 0}) = Ω̃i and + ψiN (Ωxi >0 ) = Ωxi >0 . Consider the inverse mappings ϕ−1 : Ω̃i → Ω\{xi = 0} and (+ ψiN )−1 : i Ωxi >0 → Ωxi >0 given by   xi−1 xi+1 xq x1 −1 , . . ., , xi , , . . ., , ϕi (x1 , . . . , xq ) = xi xi xi xi √ (+ ψiN )−1 (x1 , . . . , xq ) = (x1 , . . . , xi−1 , N xi , xi+1 , . . . , xq ). Lemma. We have: (1) If f ∈ W(Ωxi >0 ) then f ◦ (+ ψiN )−1 ∈ W(Ωxi >0 ). (2) If f ∈ W(Ω) then f ◦ ϕ−1 ∈ W(Ω̃i ). i Proof. Assume without loss that i = 1 and write (x1 , x2 , . . . , xq ) = (t, x̄). (1): (i) and (ii) are obvious. Since f ∈ W(Ωx1 >0 ), we have ∂1 f ∈ L1 (Ωx1 >0 ), and thus Z 1 Z Z 1 Z √ √ N N ∞> |∂1 f(t, x̄)|dtdx̄ = |∂1 f( t, x̄)||( t)′ |dtdx̄ Ωx̂1 = Z Ωx̂1 0 Z 0 Ωx̂1 1 √ N |∂t f( t, x̄)|dtdx̄ = Z 0 Ωx1 >0 |∂1 (f ◦ (+ ψ1N )−1 )(x)|dx. For the remaining partial derivatives we may argue similarly, since, for 2 ≤ i ≤ q, Z Z 1 Z 1 √ √ ′ √ 1 1 N N N |∂i f( t, x̄)|dt. |∂i f(t, x̄)|dt = |∂i f( t, x̄)||( t) |dt ≥ N 0 0 0 For (ii’) replace ∂i f with f in the last computation. (2): (i) and (ii) are again obvious. By the substitution formula, the following identities are true on all compact subsets of the complement of EΩ̃1 , where the integrands on the left-hand side are continuous and thus in L1 . Since the righthand sides are finite (without restrictions) by assumption, we indeed obtain for 2 ≤ j ≤ q and q ≥ 2: (Alternatively, use indefinite outermost integrals near 0.) Z 1Z Z 1Z 1 ∂1 f(t, x̄) dx̄dt = |∂1 f(t, x̄)||tq−1 |dx̄dt < ∞, t −1 Ω̃x̂1 −1 Ωx̂1 Z 1Z Z 1Z 1 xj |∂j f(t, x̄)||tq−2 xj |dx̄dt < ∞, ∂j f(t, x̄) 2 dx̄dt = t t −1 Ωx̂1 −1 Ω̃x̂1 Z 1Z Z 1Z 1 1 |∂j f(t, x̄)||tq−2 |dx̄dt < ∞. ∂j f(t, x̄) dx̄dt = t t −1 Ωx̂1 −1 Ω̃x̂1 Replace ∂1 f with f in the first line in order to obtain (ii’). It follows that the partial derivatives q X 1 xj 1 1 ∂j f(t, x̄) 2 , ∂1 (f ◦ ϕ−1 1 )(x) = ∂t f(t, x̄) = ∂1 f(t, x̄) − t t t t j=2 1 1 ∂j (f ◦ ϕ−1 1 )(x) = ∂j f(t, x̄) , t t are in L1 (Ω̃1 ). 2 ≤ j ≤ q,  Remark. Let Ωxi <0 := −Ωxi >0 . Similarly, one proves that if f ∈ W(Ωxi <0 ) then also f ◦ (−p ψiN )−1 ∈ W(Ωxi <0 ), for N even, where (− ψiN )−1 (x1 , . . . , xq ) = N (x1 , . . . , xi−1 , − |xi |, xi+1 , . . . , xq ), and f ◦ (+ ψiN )−1 ∈ W(Ωxi <0 ), for N odd. SMOOTH MULTIPARAMETER PERTURBATION 17 5.3. Roots of class W. Without loss we assume that the coefficients aj (where 1 ≤ j ≤ n) of P are smooth complex valued functions on 2Ω; otherwise we rescale. N ,...,Np Theorem 3.5 provides a finite collection T = {Ψ±i11 ,...,±i : 1 ≤ i1 , . . . , ip ≤ q} of p S transformations Rq , 0 → Rq , 0 such that {im(Ψ) : Ψ ∈ T } is a neighborhood of 0. N ,...,Np N ,...,Np For every Ψ±i11 ,...,±i ∈ T the family P ◦ Ψ±i11 ,...,±i admits a smooth arrangement p p N ,...,N p (for 1 ≤ j ≤ n) of its roots on 32 Ω, say; otherwise rescale. λ±i11 ,...,±i p ;j N ,...,N N N ,...,N p p Any λ±i11 ,...,±i ◦ (± ψipp )−1 ◦ ϕ−1 lies in W(Ω). By 5.2, we find λ±i11 ,...,±i ip ∈ p ;j p ;j W(Ω̃ip ), for all 1 ≤ ip ≤ q, (with the sign chosen appropriately as explained in the N ,...,Np−1 by remark). We define λ±i11 ,...,±i p−1 ;j N N ,...,N N ,...,N p−1 λ±i11 ,...,±i p−1 ;j Ω̃ip p ◦ (± ψipp )−1 ◦ ϕ−1 := λ±i11 ,...,±i ip , p ;j 1 ≤ ip ≤ q. (5.1) N ,...,N p−1 Evidently, λ±i11 ,...,±i ∈ W(Ω). After linear coordinate change if necessary, we p−1 ;j N ,...,N N ,...,N p−1 p−2 perform this procedure with λ±i11 ,...,±i and obtain λ±i11 ,...,±i ∈ W(Ω). After p−1 ;j p−2 ;j finitely many steps we end up with a parameterization λj (for 1 ≤ j ≤ n) of the roots of P on Ω such that λj ∈ W(Ω) for all j. Hence we have proved: Theorem. Let U ⊆ Rq be open. Consider a family of polynomials n X (−1)j aj (x)z n−j , P (x)(z) = z n + j=1 ∞ with coefficients aj ∈ C (U, C) (for 1 ≤ j ≤ n). Then, for each x0 ∈ U such that P is normally nonflat at x0 , there exists a neighborhood Ω ⊆ U of x0 and a parameterization λj (for 1 ≤ j ≤ n) of the roots of P on Ω such that λj ∈ W(Ω) for all j. In particular, for each λj we have ∇λj ∈ L1 (Ω).  Remarks 5.4. (1) The conclusion in theorem 5.3 is best possible: In general the roots of a smooth family of polynomials P do not allow parameterizations λj with ∇λj ∈ Lploc for any 1 < p ≤ ∞. That is shown by the example P (x)(z) = p , for 1 < p < ∞, and z n − (x1 + · · · + xq ) (for x = (x1 , . . . , xq ) ∈ Rq ) where n ≥ p−1 n ≥ 2, for p = ∞. (2) Compare this theorem with the results obtained in [7] and [8]: For a nonnegative real valued function f ∈ C k (U ), where U ⊆ Rq is open and k ≥ 2, they find that ∇(f 1/k ) ∈ L1loc (U ). Actually, for each compact K ⊆ U , one has k/(k−2) ∇(f 1/k ) ∈ Lw (K). (3) Let q√= 1. Then the proof of lemma 5.2(1) actually shows that pullback by (ψN )−1 = N preserves absolute continuity. So theorem 5.3 reproduces the result proved in [23, 4.4] (see also [26]): If q = 1 then the roots of an everywhere normally nonflat P may be arranged absolutely continuous, locally. Actually, the method we use to prove theorem 3.5 is a generalization of the method used in [23]. On an interval I the space of locally absolutely continuous functions coincides with the 1,1 Sobolev space Wloc (I). But: For q > 1, (normally nonflat) multiparameter families of polynomials do 1,1 2 not allow roots in Wloc , as the following √ example shows. Consider z = x for 2 x ∈ C = R . The roots are λ12 = ± √ x which must have a jump along some ray. The distributional derivative of x with respect to angle contains a delta distribution which is not in L1loc . If no transformation of type ϕi is needed in the algorithm (for instance if q = 1) the above reasoning provides, locally, even continuous parameterizations of the roots of P which belong to the Sobolev space W 1,1 . For: It is easy to check that 18 M. LOSIK, P.W. MICHOR, A. RAINER pullback by (+ ψiN )−1 for odd N , and by (± ψiN )−1 for even N (see remark 5.2), preserves local integrability of the distributional partial derivatives. (4) Our proof of theorem 5.3 works as well for finitely differentiable coefficients aj of P . We just have to assure that after the algorithm 3.5 we are provided with C 1 roots of P ◦ Ψ, for all Ψ ∈ T . Further, we have to replace 5.1(i) by 6.1(i’). Each time we apply lemma 3.2 we loose differentiability which we can keep track of. A minimal degree of differentiablity for the aj such that our method works depends on P . Corollary 5.5. The mapping σ : Cn → Cn from roots to coefficients (see (2.1)) admits a section which is locally in W.  6. A simpler proof of Bronshtein’s result Theorem 5.3 provides a new insight only in the non-hyperbolic case. By [5] and (alternatively) [27], the roots of a C n family of hyperbolic polynomials P of degree n may be arranged locally in a Lipschitz way. The proofs in [5] and [27] are intricate and rather sketchy (see [24] for a detailed presentation). We are able to give a very simple proof with the method developed in this paper. However, we need the coefficients of P to be in C n(n+1)/2 instead of C n . In [17] this is proved for real analytic P . 6.1. The class L. Let us denote by L(U ) the class of real valued functions with properties 5.1(ii) and (i’) f is defined and C 1 on the complement of a finite union EU of hypersurfaces in U . (iii’) The first partial derivatives of f are bounded on U \EU . 6.2. Let us use the notation of 5.2. ∈ L(Ω̃i ). Lemma. If f(x) = xi g(x) with g ∈ L(Ω) then f ◦ ϕ−1 i Proof. Assume without loss that i = 1 and write (x1 , x2, . . . , xq ) = (t, x̄). The only thing to check is (iii’). For 2 ≤ j ≤ q we find that 1 1 1 ∂j (f ◦ ϕ−1 1 )(x) = ∂j f(t, x̄) = ∂j g(t, x̄) t t t is bounded on Ω̃1 \EΩ̃1 , since g ∈ L(Ω). It follows that also q X 1 xj 1 1 ∂j f(t, x̄) 2 ∂1 (f ◦ ϕ1−1 )(x) = ∂t f(t, x̄) = ∂1 f(t, x̄) − t t t t j=2 is bounded on Ω̃1 \EΩ̃1 , since |xj | < |t| for all 2 ≤ j ≤ q.  6.3. The following lemma is used in 6.5 in order to estimate the total loss of differentiability resulting from the iterated application of the reduction procedure used in our algorithm. Lemma. For n ∈ N let R(n) denote the family of all rooted trees T with vertices labeled in the following way: the root is labeled n, the labels of the successors of a vertex labeled m form a partition of m, the leaves (vertices with no successors) are all labeled 1. Define d(n) := maxT ∈R(n) {sum over all labels ≥ 2 in T }. Then d(n) = 21 n(n + 1) − 1. Proof. Observe that d(1) = 0. Then it suffices to show that d(n) = n + d(n − 1) for n ≥ 2. To this end we show (using induction on n) that d(n) ≥ d(n1 ) + · · · + d(np ) SMOOTH MULTIPARAMETER PERTURBATION 19 for n1 + · · · + np = n + 1, where p ≥ 2 and ni ∈ N. By induction hypothesis, this inequality is equivalent to 1 1 1 n(n + 1) − 1 ≥ n1 (n1 + 1) + · · · + np (np + 1) − p 2 2 2 which is easily verified.  5 3 A rooted tree in R(5): 1 1 1 1 1 6.4. Let U ⊆ Rq be an open neighborhood of 0. Consider a family of hyperbolic polynomials n X (−1)j aj (x)z n−j P (x)(z) := z n + j=1 m with coefficients aj ∈ C (U ) (for 1 ≤ j ≤ n). For an open neighborhood I ⊆ R of 0 we define ext(P )(x, t)(z) := P (x)(z) + t∂z P (x)(z), Its coefficients x ∈ U, t ∈ I. a1 (ext(P ))(x, t) = a1 (P )(x) − nt, (6.1) aj (ext(P ))(x, t) = aj (P )(x) − (n − j + 1)taj−1 (P )(x), m 2 ≤ j ≤ n, belong to C (U × I). The family ext(P ) is hyperbolic as well by the following lemma. Pn Lemma. If P (z) = z n + j=1 (−1)j aj z n−j is hyperbolic then so is P (z) + tP ′ (z), for t ∈ R. Proof. Let λ1 , . . . , λn denote the roots of P . Then n  X P (z) + tP ′ (z) = P (z) 1 + t j=1 1  . z − λj Suppose that there is a µ with Im(µ) 6= 0 and P (µ) + tP ′ (µ) = 0. It implies n n  1  X X 1 Im 0= = − Im(µ) , µ − λj |µ − λj |2 j=1 j=1 a contradiction.  6.5. Roots of class L and C 0,1. Consider a family of hyperbolic polynomials n X n (−1)j aj (x)z n−j P (x)(z) = z + j=1 with coefficients aj (for 1 ≤ j ≤ n) of class C n(n+1)/2 on an open neighborhood of 0 ∈ Rq . Let us modify the argumentation in 5.3. Perform step (1) of the algorithm in 3.5 for P . In step (2) we split P (x) = P1 (x) · · · Pl (x) and then replace each factor Ph (for 1 ≤ h ≤ l) by ext(Ph ). After removing fixed points (maintaining the notation ext Ph ), we apply step (3) to Q h ext(Ph ). For simplicity let us suppress the index h, i.e., we treat just one factor ext(P ). It is easy to deduce from (6.1) that each ak (ext(P )) (for 2 ≤ k ≤ n) 20 M. LOSIK, P.W. MICHOR, A. RAINER i vanishes of order ≤ k − 1. So the coefficients of the reduced family ext(P )A◦ϕ (M ) in (3.5) belong to C n(n+1)/2−n near 0 ∈ Rq × R. Note that here we used the finite i differentiability version of lemma 3.2 and 3.3. Since not all roots of ext(P )A◦ϕ (M ) (0) coincide, we may continue in step (1) or (2). After m ≤ n−1 applications of that procedure (we can split at most n−1 times) we obtain a finite collection P = {Pi1 ,...,im : 1 ≤ ij ≤ q + j, 1 ≤ j ≤ m} of families of hyperbolic polynomials defined for (x, t) near 0 ∈ Rq × Rm of the form Pi1 ,...,im = ext(· · · (ext(ext(P ) ◦ A1 ◦ ϕi1 ) ◦ A2 ◦ ϕi2 ) ◦ · · · ) ◦ Am ◦ ϕim , where the Aj denote linear coordinate changes. By construction, each Pi1 ,...,im ∈ P admits C 1 parameterizations λi1 ,...,im ;j (for 1 ≤ j ≤ n) of its roots: The total loss of differentiability occurring in the algorithm is ≤ d(n) and, by lemma 6.3, we have 1 2 n(n + 1) − d(n) = 1. Note that we introduced m new parameters t = (t1 , . . . , tm ). We write Ωr for the cube Ω in Rr , introduced in 5.2 (then Ωq = Ω). Note S that Ωq+j = {ϕij (Ωq+j ) : 1 ≤ ij ≤ q + j} for each 1 ≤ j ≤ m. We can suppose that any λi1 ,...,im ;j ∈ L(Ωq+m ). By construction (step (3) of the algorithm), we may assume without loss that λi1 ,...,im ;j (y) = yim µi1 ,...,im ;j (y) with µi1 ,...,im ;j ∈ L(Ωq+m ), where y = (x, t) ∈ Rq × Rm . By lemma 6.2, we find λi1 ,...,im ;j ◦ ϕ−1 im ∈ ext q+m ), for all 1 ≤ i ≤ q + m. We define λ ∈ L(Ω ) by L(Ω̃q+m m i1 ,...,im−1 ;j im λext i1 ,...,im−1 ;j Ω̃q+m im := λi1 ,...,im ;j ◦ ϕi−1 , m 1 ≤ im ≤ q + m. Then we restrict to the hyperplane H = {tm = 0} with respect to the coordinates before the change Am , i.e., we define λi1 ,...,im−1 ;j := λext i1 ,...,im−1 ;j |H . It preserves the class L: without loss the intersection of H with any hyperplane in the complement S q+m : 1 ≤ i ≤ q + m} in Ω and with any hypersurface in EΩq+m of {Ω̃q+m m im has measure zero (if necessary we modify the linear coordinate change Am slightly which is guaranteed by lemma 3.2). So Ωq+m ∩ H = Ωq+m−1 and λi1 ,...,im−1 ;j ∈ L(Ωq+m−1 ). We repeat the procedure with λi1 ,...,im−1 ;j . After m steps we obtain a parameterization λj (for 1 ≤ j ≤ n) of the roots of P on Ωq = Ω such that λj ∈ L(Ωq ) for all j. Let µj be any continuous parameterization of the roots of P on Ω (e.g. by (±) ordering the roots increasingly). Then all one-sided partial derivatives ∂k µj of µj exist everywhere in Ω: this follows from the one parameter case (e.g. [5] or [1]) since (±) (±) (±) ∂k µj (x) = ∂t |0 µj (x + tek ). We find that all ∂k µj are bounded by M , say, in (±) (±) Ω\EΩ , since ∂k µj (x) (for 1 ≤ j ≤ n) differ from ∂k λj (x) (for 1 ≤ j ≤ n) by just a permutation whenever the latter exist (see below). Let x0 ∈ EΩ . Changing basis if necessary, we can assume that all coordinate directions are transversal to EΩ , (±) (±) locally. The mean value theorem implies that ∂k µj (x0 ) = ∂t |0 µj (x0 +tek ) ≤ M , 0,1 too. Thus we have proved that, locally, each µj ∈ C (Ω). We prove the claim: It suffices to consider the one parameter case. Let ν 1 = 1 (ν1 , . . . , νn1 ) and ν 2 = (ν12 , . . . , νn2 ) be parameterizations of the roots of a polynomial P depending on a parameter t near 0 ∈ R and assume that the right-sided (say) derivatives (ν 1 )(+) (0) and (ν 2 )(+) (0) exist. After using the splitting lemma 2.2 and removing fixed points, we find, for 1 ≤ i ≤ n and j = 1, 2,    j  ν j (t) ν (t) σi ((ν j )(+) (0)) = σi lim = lim σi = lim t−i ai (t). tց0 tց0 tց0 t t So (ν 1 )(+) (0) and (ν 2 )(+) (0) represent different arrangements of the roots of the same polynomial with coefficients limtց0 t−i ai (t) (for 1 ≤ i ≤ n). Hence the claim. Therefore, we have proved: SMOOTH MULTIPARAMETER PERTURBATION 21 Theorem. Let U ⊆ Rq be open. Consider a family of hyperbolic polynomials n X (−1)j aj (x)z n−j , P (x)(z) = z n + j=1 n(n+1)/2 with coefficients aj ∈ C (U ) (for 1 ≤ j ≤ n). Then, for each x0 ∈ U , there exists a neighborhood Ω ⊆ U of x0 and a parameterization λj (for 1 ≤ j ≤ n) of the roots of P on Ω such that λj ∈ L(Ω) for all j. Moreover, any continuous parameterization of the roots of P is locally Lipschitz continuous on U .  Remark. If r is the maximal multiplicity of the roots of P (x), then we need only C r(r+1)/2 coefficients aj of P for our method to work. Nevertheless, Bronshtein [5] and Wakabayashi [27] prove local Lipschitz continuity of the roots for C r coefficients. 7. Smooth perturbation of normal operators 7.1. Perturbation of normal matrices. A smooth family of normal complex n×n matrices A(x) = (Aij (x))1≤i,j≤n, where x varies in a suitable parameter space (a neighborhood U of 0 ∈ Rq or a connected smooth manifold M ), is called normally nonflat at x0 , if χA(x) is normally nonflat at x0 , where χA(x) (λ) = det(A(x) − λI) is the characteristic polynomial of A(x). Theorem. Let A(x) = (Aij (x))1≤i,j≤n be a family of normal complex matrices (acting on a complex vector space V = Cn ). Then: (1) Suppose that Aij : Rq , 0 → C (for 1 ≤ i, j ≤ n) are germs at 0 of smooth functions, and that A(x) is normally nonflat at 0. Then there exists a finite S N ,...,Np collection T = {Ψ±i11 ,...,±i : 1 ≤ i1 , . . . , ip ≤ q} such that {im(Ψ) : Ψ ∈ p T } is a neighborhood of 0 and for each Ψ ∈ T the family A ◦ Ψ allows a smooth parameterization of its eigenvalues and eigenvectors near 0. (2) If A(x) is a smooth family of Hermitian matrices which is defined and normally nonflat at any x in a connected smooth manifold M , then there exists a smooth manifold M ′ and a surjective smooth projection Φ : M ′ → M which is a locally finite composition of blow-ups centered at single points such that A ◦ Φ allows smooth eigenvalues and eigenvectors on M ′ , locally. If π : M̃ ′ → M ′ is the universal covering of M ′ , we find smooth eigenvalues of A ◦ Φ ◦ π on M̃ ′ , globally. (3) Suppose that U ⊆ Rq is a neighborhood of 0, the Aij : U → C (for 1 ≤ i, j ≤ n) are smooth functions and A(x) is normally nonflat at 0. For some neighborhood Ω ⊆ Rq of 0 there exist parameterizations of the eigenvalues and eigenvectors of A in W(Ω) (resp. W(Ω, Cn )). If A is normally nonflat at any x ∈ U , then A allows eigenvalues and eigenvectors with first order partial derivatives in L1loc (U ) (resp. L1loc (U, Cn )). Proof. We adapt the proof of [1, 7.6]; see also [23, 7.1]. (1) By theorem 3.5, for the characteristic polynomial χA(x) (λ) = det(A(x) − λI) = n X j=0 Trace(Λj A(x))λn−j = λn + n X (−1)j aj (x)λn−j j=1 (7.1) there exists a collection T0 of the form indicated in the theorem such that for each Ψ0 ∈ T0 the family χA◦Ψ0 admits a smooth parameterization µ1 , . . . , µn of its roots (the eigenvalues of A ◦ Ψ0 ) near 0. Consider the following algorithm: (a) Not all eigenvalues of A(0) agree. Let ν1 , . . . , νl denote the pairwise distinct eigenvalues of A(0) with respective multiplicities m1 , . . . , ml . Assume without loss 22 M. LOSIK, P.W. MICHOR, A. RAINER that ν1 = µ1 (0) = · · · = µm1 (0), ν2 = µm1 +1 (0) = · · · = µm1 +m2 (0), · · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · · νl = µn−ml +1 (0) = · · · = µn (0). This defines a partition into subsets of smooth eigenvalues such that, for x near 0, they do not meet each other if they belong to different subsets. For 1 ≤ h ≤ l consider M ker(A(Ψ0 (x)) − µi (x)) Vx(h) := {i : νh =µi (0)} = ker ◦{i : νh =µi (0)}
(A(Ψ0 (x)) − µi (x)) . Note that the order of the compositions in the above expression is not relevant. So (h) Vx is the kernel of a smooth vector bundle homomorphism B(x) of constant rank, and thus is a smooth vector subbundle of the trivial bundle Rq × V → Rq . This can be seen as follows: Choose a basis of V such that A(Ψ0 (0)) is diagonal. By the elimination procedure one can construct a basis for the kernel of B(0). For x near 0, the elimination procedure (with the same choices) gives then a basis of the kernel of B(x). The elements of this basis are then smooth in x near 0. It follows that it suffices to find smooth eigenvectors in each subbundle V (h) simultaneously (in the sense explained in the proof of theorem 3.5), expanded in the constructed smooth frame field. But in this frame field the vector subbundle looks again like a constant vector space. So we feed each of these parts (A ◦ Ψ0 restricted to V (h) , as matrix with respect to the frame field) into step (b) below. Analogously to 3.5, step (b) is performed for all 1 ≤ h ≤ l simultaneously; but here we suppress the index h. (b) All eigenvalues of A(0) coincide and equal a1 (0)/n, according to (7.1). Eigenvectors of A(x) are also eigenvectors of A(x)−(a1 (x)/n)I, thus we may replace A(x) by A(x)−(a1 (x)/n)I and assume that the first coefficient of the characteristic polynomial (7.1) vanishes identically. Then A(0) = 0. If A(x) = 0 for x near 0 we choose the eigenvectors constant. Otherwise, by assumption, not all Aij are infinitely flat at 0. By lemma 3.2, there exists a linear coordinate change T on Rq , 0, positive integers mij , and smooth germs Bij,k , for 1 ≤ i, j ≤ n and 1 ≤ k ≤ q, such that m (Aij ◦ T ◦ ϕk )(x) = xk ij Bij,k (x), where Bij,k (0) 6= 0 if and only if Aij is not infinitely flat at 0. Let m := min{mij : Bij,k (0) 6= 0} and assume (without restriction) that mij ≥ m for the flat Aij as well. Then T ◦ϕk (Aij ◦ T ◦ ϕk )(x) = xm k Aij,(m) (x), ◦ϕk , where 1 ≤ i, j ≤ n and 1 ≤ k ≤ q. It follows from (7.1) for smooth germs ATij,(m) ◦ϕk ◦ϕk (x))1≤i,j≤n has the form (x) = (ATij,(m) that the characteristic polynomial of AT(m) χAT ◦ϕk (x) (λ) = λn + (m) n X (−1)j x−mj aj (T (ϕk (x)))λn−j . k j=2 By theorem 3.5, for each 1 ≤ k ≤ q there exists a collection T1,k of the form indicated in the theorem such that for each Ψ1,k ∈ T1,k the family χAT ◦ϕk ◦Ψ (m) 1,k ◦ϕk ◦ Ψ1,k ) near 0. admits smooth parameterizations of its roots (eigenvalues of AT(m) ◦ϕk ◦Ψ1,k )(x) are also eigenvectors of (A◦T ◦ϕk ◦Ψ1,k )(x). There Eigenvectors of (AT(m) SMOOTH MULTIPARAMETER PERTURBATION 23 exist 1 ≤ i, j ≤ n (for instance those with mij = m and Bij,k (0) 6= 0) such that ◦ϕk ◦ϕk ◦ϕk (Ψ1,k (0)) (0) 6= 0 and thus not all eigenvalues of AT(m) (Ψ1,k (0)) = ATij,(m) ATij,(m) ◦ϕk ◦ Ψ1,k , for 1 ≤ k ≤ q and Ψ1,k ∈ T1,k , into (a). are equal. Feed AT(m) The algorithm stops after finitely many steps and produces the required finite collection T . (2) The statement for the eigenvalues is an immediate consequence of theorem 4.2. In view of the previous algorithm, which in the Hermitian case does not use transformations of type ± ψiN , we may adapt the proof of theorem 4.2 in order to show that there exists the required Φ : M ′ → M such that A ◦ Φ allows smooth eigenvectors, locally. (3) The assertion for the eigenvalues follows immediately from theorem 5.3 (applied to (7.1)). For the eigenvectors the statement follows from (1) in an analogous way as theorem 5.3 follows from theorem 3.5 (see section 5).  Remark. Simple examples show that neither the condition that the matrices A(x) are normal nor normal nonflatness can be omitted in (the eigenvector part of) theorem 7.1: For x ∈ R consider     1 cos x2 sin x2 0 1 A1 (x) := , A2 (0) := 0. , and A2 (x) := e− x2 x 0 sin x2 − cos x2 Any choice of eigenvectors of A1 has a pole at 0. The eigenvectors of A2 cannot be chosen continuously near 0. 7.2. Perturbation of unbounded normal operators. Let x 7→ A(x) be a smooth family of unbounded normal (resp. selfadjoint) operators in a Hilbert space with common domain of definition and with compact resolvents, defined for x near 0 ∈ Rq . That means: There is a dense subspace V of the Hilbert space H such that V is the domain of definition of each A(x), and such that each A(x) is closed and A(x)∗ A(x) = A(x)A(x)∗ (resp. A(x) = A(x)∗ ), where the adjoint operator A(x)∗ is defined as usual by hA(x)u, vi = hu, A(x)∗ vi for all v for which the left-hand side is bounded as function in u ∈ H. Note that the domain of definition of A(x)∗ is V . Moreover, we require that x 7→ hA(x)u, vi is smooth for each u ∈ V and v ∈ H. This implies that x 7→ A(x)u is of the same class Rq → H for each u ∈ V , by [14, 2.3] or [10, 2.6.2]. Recall the resolvent lemma in [15] (see also [1]): If A(x) is smooth, then also the resolvent (A(x) − z)−1 is smooth into L(H, H) in x and z jointly. Note that the multidimensional version we need here can be proved analogously. Let z0 be an eigenvalue of A(0) of multiplicity n. Choose a simple closed curve γ in the resolvent set of A(0) enclosing only z0 among all eigenvalues of A(0). Since the global resolvent set {(x, z) ∈ Rq × C : (A(x) − z) : V → H is invertible} is open, no eigenvalue of A(x) lies on γ, for x near 0. Consider Z 1 (A(x) − z)−1 dz =: P (x, z0), x 7→ − 2πi γ a smooth family of projections (on the direct sum of all eigenspaces corresponding to eigenvalues in the interior of γ) with finite dimensional ranges and constant ranks (see [1] or [15]). So for x near 0, there are equally many eigenvalues in the interior of γ. Theorem. Let x 7→ A(x) be a smooth family of unbounded normal operators in a Hilbert space with common domain of definition and with compact resolvents, defined for x near 0 ∈ Rq . Suppose that z0 is an n-fold eigenvalue of A(0) such that x 7→ P (x, z0 )A(x)|P (x,z0 )(H) is normally nonflat at 0. Then: 24 M. LOSIK, P.W. MICHOR, A. RAINER N ,...,N p (1) There exists a finite collection T = {Ψ±i11,...,±i : 1 ≤ i1 , . . . , ip ≤ q} such p S that {im(Ψ) : Ψ ∈ T } is a neighborhood of 0, and, for each Ψ ∈ T , near 0 the n eigenvalues of A ◦ Ψ in a neighborhood of z0 and the corresponding eigenvectors allow smooth parameterizations. (2) Let A(x) be a family of selfadjoint operators. Then there exists an open neighborhood U ⊆ Rq of 0, a smooth manifold U ′ , and a surjective smooth projection Φ : U ′ → U which is a locally finite sequence of blow-ups centered at single points such that the n eigenvalues near z0 of A ◦ Φ and the corresponding eigenvectors admit locally smooth parameterizations on U ′ . (3) There exists a neighborhood Ω ⊆ Rq of 0 such that the n eigenvalues of A near z0 and the corresponding eigenvectors may be chosen in W(Ω) (resp. W(Ω, Cn )). Proof. Let γ and P (x, z0) be as above. Then for x near 0, there are equally many eigenvalues in the interior of γ. Let us call them λi (x) for 1 ≤ i ≤ n (repeated with multiplicity) and let us denote by ei (x) for 1 ≤ i ≤ n a corresponding system of eigenvectors of A(x). By the residue theorem we have Z n X 1 p z p (A(x) − z)−1 dz λi (x) ei (x)hei (x), i = − 2πi γ i=1 which is smooth in x near 0, as a family of operators in L(H, H) of rank n. We use (the multidimensional version of) claim 2 from [1, 7.8] (with the same proof): Let x 7→ T (x) ∈ L(H, H) be a smooth family of operators of rank n in Hilbert space such that T (0)T (0)(H) = T (0)(H). Then x 7→ Trace(T (x)) is smooth near 0. So the Newton polynomials Z n X 1 Trace z p (A(x) − z)−1 dz, λi (x)p = − 2πi γ i=1 are smooth for x near 0, and thus also the elementary symmetric functions X λi1 (x) · · · λip (x). i1 <···<ip It follows that {λi (x) : 1 ≤ i ≤ n} represents the set of roots of a polynomial of degree n with smooth coefficients. 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Vol. XLV (1999), 327–337. [27] S. Wakabayashi, Remarks on hyperbolic polynomials, Tsukuba J. Math. 10 (1986), 17–28. Mark Losik: Saratov State University, Astrakhanskaya, 83, 410026 Saratov, Russia E-mail address: [email protected] Peter W. Michor: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria; and: Erwin Schrödinger Institut für Mathematische Physik, Boltzmanngasse 9, A-1090 Wien, Austria E-mail address: [email protected] Armin Rainer: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria E-mail address: [email protected]