Quasianalytic perturbation of multi-parameter hyperbolic polynomials

and symmetric matrices

Krzysztof Jan Nowak

IMUJ PREPRINT 2010/02

Abstract

This paper investigates hyperbolic polynomials with quasianalytic coefficients. Our main purpose is to prove factorization theorems for such polynomials, and next to generalize the results of K. Kurdyka and L. Paunescu about perturbation of analytic families of symmetric matrices to the quasianalytic settings.

Generally, the perturbation problem concerns the issue whether, given a family of monic polynomials with coefficients from a certain class of functions, one can represent its roots as function of this class. Hyperbolic polynomials with analytic coefficients in one variable were studied by Rellich [24, 25], which was linked with his investigation into the behaviour of eigenvalues of symmetric matrices under one-parameter analytic perturbation. This oneparameter theory, initiated by Rellich, culminated in the work of Kato [8].

2010 MSC: 14P15, 32B20, 15A18, 26E10.

Key words: quasianalytic perturbation, hyperbolic polynomials, quasianalytic and arc-quasianalytic functions, polynomially bounded structures, eigenvalues, eigenspaces, symmetric and antisymmetric matrices, spectral theorem, quasianalytic diagonalization.

One-parameter families of hyperbolic polynomials were contemporarily studied in [2, 10], as well. Recently, Kurdyka–Paunescu [12] developed multiparameter analytic perturbation theory. Our purpose is to carry over this multi-parameter theory to the quasianalytic settings. This task is also the subject of a contemporaneous paper [23].

The main purpose of this paper is to establish certain splitting theorems for quasiordinary hyperbolic polynomials with quasianalytic coefficients. Our main results are Theorem 1 and Theorem 1^{∗} to the effect that every such polynomial splits into linear factors with coefficients which are arc-Q-analytic functions or smooth, definable in the real field with restricted Q-analytic functions, respectively. In order to prove Theorem 1, we introduce the concepts of arc-symmetric Q-sets and a noetherian arc-symmetric topology, which correspond to those by Kurdyka [11] for semialgebraic sets. Also, we use global (canonical) desingularization results by Bierstone–Milman [4, 5]. Theorem 1^{∗} is derived from Theorem 1 by means of a description of arc-Q-analytic functions due to Bierstone–Milman–Valette [7] and a quasianalytic version of Glaeser’s composite function theorem presented in our paper [21] (wherein we demonstrate how to carry over the results of Bierstone–Milman– Pawłucki [6] to the quasianalytic settings). Finally, we draw some applications to perturbation of symmetric matrices with quasianalytic entries.

In our earlier approach to quasianalytic families of hyperbolic polynomials and symmetric matrices (cf. [17]), we applied a generalization of the Abhyankar–Jung theorem for henselian k[x]-algebras of formal power series, which are closed under reciprocal, power substitution and division by a coordinate, given in our paper [16]. This allowed us to carry over that theorem to the local rings of quasianalytic function germs in several variables in polynomially bounded o-minimal structures. Our proof of that theorem made use, however, of Luengo’s statement that every quasiordinary Weierstrass polynomial in the Tschirnhausen form is ν-quasiordinary in the sense of Hironaka. Therefore, those results of ours bear a relative character, because it turned out, as indicated in [9], that Luengo’s proof seems to have an essential gap.

As in our previous papers [14, 15], we fix a family Q = (Q_{m})_{m∈ℕ} of sheaves of local ℝ-algebras of smooth functions on ℝ^{m}. For each open subset U ⊂ ℝ^{m}, Q(U) = Q_{m}(U) is thus a subalgebra of the algebra C_{m}^{∞}(U) of real smooth functions on U. By a Q-analytic function (or a Q-function, for abbreviation) we mean any function f ∈ Q(U). Similarly,

f = (f_{1}, …, f_{k}) : U ⟶ ℝ^{k}

is called a Q-analytic mapping (or a Q-mapping) if so are its components f_{1}, …, f_{k}. We impose on this family of sheaves the following six conditions:

1. each algebra Q(U) contains the restrictions of polynomials;

2. Q is closed under composition, i.e. the composition of Q-mappings is a Q-mapping (whenever it is well defined);

3. Q is closed under inverse, i.e. if φ : U ⟶ V is a Q-mapping between open subsets U, V ⊂ ℝ^{m}, a ∈ U, b ∈ V and if ∂φ/∂x(a) ≠ 0, then there are neighbourhoods U_{a} and V_{b} of a and b, respectively, and a Qdiffeomorphism ψ : V_{b} ⟶ U_{a} such that φ ∘ ψ is the identity mapping on V_{b};

4. Q is closed under differentiation;

5. Q is closed under division by a coordinate, i.e. if f ∈ Q(U) and f(x_{1}, …, x_{i−1}, a_{i}, x_{i+1}, …, x_{m}) = 0 as a function in the variables x_{j}, j ≠ i, then f(x) = (x_{i} − a_{i})g(x) with some g ∈ Q(U);

6. Q is quasianalytic, i.e. if f ∈ Q(U) and the Taylor series f̂_{a} of f at a point a ∈ U vanishes, then f vanishes in the vicinity of a.

We have not attempted to avoid some possible redundancy among them; for instance, it is easy to check, via the Taylor formula, that condition 5 implies condition 4.

By means of Q-mappings, one can build, in the ordinary manner, the category Q of Q-manifolds and Q-mappings, which is a subcategory of that of smooth manifolds and smooth mappings. Also, the category Q gives rise to the geometry of Q-semianalytic and Q-subanalytic sets. What is unavailable in the Q-analytic geometry are the Weierstrass preparation and division theorems, whereon majority of the classical analytic geometry relies. Instead, it is largely based on transformation to normal crossings by blowing up along smooth centers and desingularization algorithms, which are ensured by the above conditions (as proven in papers [4, 5, 26]).

Often it is convenient to use the terminology of o-minimal structures. Denote by R_{Q} the expansion of ℝ by restricted Q-functions, which is an ominimal polynomially bounded structure with exponent field ℚ and admits smooth quasianalytic cell decomposition (cf. [26, 14]. From now on, the word ”definable” means ”definable (with parameters) in the structure R_{Q}”.

Let Ω ⊂ ℝ^{m} be an open subset. We call a monic polynomial

f(x; t) = t^{n} + a_{n−1}(x)t^{n−1} + ⋯ + a_{0}(x) ∈ Q(Ω)[t]

hyperbolic if, for each value of the parameters x ∈ Ω, all its roots are real. This is an abbreviated name for ”a quasianalytic family of hyperbolic polynomials”. The following result refers to one-parameter hyperbolic polynomials.

Proposition 1. Every hyperbolic polynomial

f(x; t) = t^{n} + a_{n−1}(x)t^{n−1} + ⋯ + a_{0}(x) ∈ Q(c, d)[t]

with coefficients quasianalytic on an interval (c, d) ⊂ ℝ, splits into linear factors over Q(c, d).

The problem being local, we can regard the coefficients of the polynomial as Q-analytic germs at 0 ∈ ℝ. The counterpart of the Newton–Puiseux theorem holds for the local ring Q_{1} of Q-analytic germs at 0 ∈ ℝ, because it is a discrete valuation ring (cf. [18], Remark 3). Therefore all roots of f(x; t) are of the form

∑_{k=0}^{r−1} α_{k}(x) · x^{k/r}, where α_{k}(x) ∈ Q_{1}, r = n!

(loc. cit., Remark 2). But all algebraic conjugates

∑_{k=0}^{r−1} α_{k}(x) · (ϵ^{i}x^{1/r})^{k}, i = 0, …, r − 1,

where ϵ is a primitive r-th root of unity, are roots of f(x; t) too. Since all the roots are real, it follows that

α_{k}(x) ≡ 0 for all k = 1, …, r − 1,

as required.

Let M be a Q-manifold. We call a function g : M ⟶ ℝ arc-quasianalytic (or arc-Q-analytic) if, for every Q-analytic arc γ : (−δ, δ) ⟶ M, the superposition g ∘ γ is Q-analytic. It is well known that every arc-Q-function is continuous (cf. [11, 3, 20]).

Now, consider a monic polynomial

f(x; t) = t^{n} + a_{n−1}(x)t^{n−1} + ⋯ + a_{0}(x) ∈ Q(Ω)[t]

with quasianalytic coefficients on an open subset Ω ⊂ ℝ^{m}. We call f(x; t) a quasiordinary polynomial if its discriminant D(x) is a normal crossing on Ω. Our main purpose is to establish a multi-parameter counterpart of the foregoing result, namely, the following theorem on splitting a hyperbolic quasiordinary polynomial into linear factors over the ring of arc-Q-analytic functions:

Theorem 1. (Splitting of hyperbolic quasiordinary polynomials) Let Ω be an open, simply connected subset of ℝ^{m}. Then every hyperbolic quasiordinary polynomial

f(x; t) = t^{n} + a_{n−1}(x)t^{n−1} + ⋯ + a_{0}(x) ∈ Q(Ω)[t]

splits into linear factors of the form

f(x; t) = ∏_{i=1}^{n}(t − ψ_{i}(x)), x ∈ Ω,

where ψ_{i}(x) are arc-Q-functions on Ω. Furthermore, the functions ψ_{i}(x), i = 1, …, n, enjoy at every point P ∈ Ω the following property (T):

(T) there exist formal power series Ψ_{i}, i = 1, …, n, centered at P such that for each Q-analytic arc γ(τ) through P, γ(0) = P, not contained in the zero locus of the discriminant D(x) of f(x; t), the Taylor series of ψ_{i} ∘ γ at zero are Ψ_{i} ∘ γ̂, i = 1, …, n; here γ̂ denotes the Taylor series of γ at zero.

Before proceeding with the proof of this theorem, we shall introduce some terminology and results indispensable for our approach. Let M be a definable, locally closed, smooth submanifold of an affine space ℝ^{N}. By a Q-leaf in M we mean a definable, connected, locally closed subset that is a smooth submanifold of M. We say that a definable subset E ⊂ M is arc-symmetric when one of the two equivalent conditions holds:

i) if γ : (−1, 1) ⟶ M is a Q-analytic arc such that γ(−1, 0) ⊂ E, then γ(−1, 1) ⊂ E;

ii) if γ : (−1, 1) ⟶ M is a Q-analytic arc such that Int (γ^{−1}(E)) ≠ ∅, then γ(−1, 1) ⊂ E.

We shall call E an arc-symmetric Q-subset of M. This is an o-minimal generalization of the concept of an arc-symmetric semialgebraic subset, introduced by Kurdyka [11] and inspired by his joint article [13]. By the curve selection lemma, every arc symmetric Q-subset E of M is closed.

Applying smooth definable stratifications (and even the decompositions into Q-leaves instead) and the global (canonical) desingularization of quasianalytic hypersurfaces due to Bierstone–Milman [4, 5], we are able to carry over Kurdyka’s results about arc-symmetric semialgebraic sets to the case of arc-symmetric Q-sets. Below, we state three quasianalytic counterparts of his Theorem 1.4, Theorem 2.6 and Corollary 2.8, whose proofs can be repeated almost verbatim.

Proposition 2. There exists a unique noetherian topology AS on M such that its closed sets are precisely the arc-symmetric Q-subsets of M.

This follows, via induction on the dimension of the ambient space M, from the decomposition of a definable set into finitely many Q-leaves and the

Observation. Let E ⊂ M be an arc-symmetric Q-subset and Γ ⊂ M a Q-leaf. If dim(Γ ∩ R) = dim Γ, then Γ ⊂ E.

The topology described above shall be called the arc-symmetric topology.

Theorem 2. Let Y be a reduced Q-analytic hypersurface in ℝ^{m} of dimension m−1. Further, let U be an open, definable, relatively compact subset of ℝ^{m}, X := Y ∩ U and E an arc-symmetric Q-subset of X of dimension m−1. Consider a (canonical) desingularization σ : X̃ ⟶ X which is a finite sequence of global blowings-up with smooth centers. Then E is irreducible in the arc-symmetric topology iff there is a (unique) connected component C of X̃ such that σ(C) coincides with the closure (in the ordinary topology) of the set Reg_{m−1}(E) of those points at which E is smooth of dimension m − 1.

Kurdyka’s proof applies directly to the above quasianalytic version of his Theorem 2.6, because the image σ(E) of the exceptional divisor E of σ is an arc-symmetric Q-subset of dimension < (m − 1).

Corollary 2. Under the foregoing assumptions, if E is an arc-symmetric Q-subset of X of dimension m − 1, then the two conditions are equivalent:

i) E is irreducible in the arc-symmetric topology;

ii) each two points of the closure \overline{Reg_{m−1}(E)} of Reg_{m−1}(E) can be joined by a Q-analytic arc lying in \overline{Reg_{m−1}(E)}.

Remark 1. In the proof of this corollary, Kurdyka’s argument why any given two points P, Q on a connected real analytic submanifold V in a real affine space ℝ^{N} can be joined by an analytic arc (by means of geodesics) should be replaced in the quasianalytic settings by two observations. First, the problem reduces to the case of a connected open subset of ℝ^{N} via a tubular neighbourhood (in the category of Q-manifolds) (W, ϱ) of the Qsubmanifold V under study, where ϱ : W ⟶ V is a retraction. Indeed, one must compose a Q-analytic arc in W with the retraction ϱ. Next, take a broken line γ : [0, 1] ⟶ W, γ(0) = P, γ(1) = Q, which lies in W and joins the two points P = (P_{1}, …, P_{N}), Q = (Q_{1}, …, Q_{N}) ∈ V ⊂ ℝ^{N}. Then, applying a variant of the Weierstrass approximation theorem, stated below, it suffices to approximate the components γ_{i}(t), i = 1, …, N, of γ by real polynomials p_{i}(t) in one variable t such that p_{i}(0) = P_{i}, p_{i}(1) = Q_{i} for i = 1, …, N.

Lemma 1. Consider a continuous real function f(t) on a compact interval [c, d] and a finite number of points ξ_{1}, …, ξ_{s} ∈ [c, d]. Then for each real number ε > 0, there exists a polynomial p(t) such that

|f(t) − p(t)| < ε for all t ∈ [c, d] and f(ξ_{1}) = p(ξ_{1}) , …, f(ξ_{s}) = p(ξ_{s}).

This lemma is a combination of Weierstrass approximation and Lagrange interpolation with nodes ξ_{1}, …, ξ_{s}.

At this stage we can readily turn to the proof of our main result.

Proof of Theorem 1. The zero locus

Y := {(x, t) ∈ Ω × ℝ_{t} : f(x; t) = 0} ⊂ ℝ_{x}^{m} × ℝ_{t}

is a reduced hypersurface of codimension 1. Due to the identity principle for arc-Q-functions, the splitting problem is local. We may thus confine our proof to an open, relatively compact subset

U = (−δ, δ)^{m} × (−M, M)

of Ω × ℝ which contains all roots of f(x; t) over (−δ, δ)^{m}, and such that the discriminant of f(x, t) is of the form D(x) = x^{α} · u(x), where α ∈ ℕ^{m} and u(x) is a nowhere vanishing, Q-analytic function on (−δ, δ)^{m}. In the sequel, we shall apply Theorem 2 to the trace X := Y ∩ U.

The set {x ∈ ( δ, δ)^{m} : D(x) ≠ 0} contains the union of 2^{m} open cubes Q_{j} cut out from U by the cross x_{1} · . . . · x_{m} = 0. Obviously, the set X over each cube Q_{j} is the union of the graphs of n Q-analytic and Q-subanalytic functions ψ_{j,1}(x), …, ψ_{j,n}(x) on Q_{j} such that

ψ_{j,1}(x) < … < ψ_{j,n}(x) for all x ∈ Q_{j}.

By the classical Abhyankar–Jung theorem (see e.g. [1, 9, 18]), the formal roots Ψ_{1}(x), …, Ψ_{n}(x) of f(x; t) at 0 ∈ ℝ^{m} belong to

Ψ_{1}(x), …, Ψ_{n}(x) ∈ ℂ[[x^{1/r}]], x^{1/r} = (x_{1}^{1/r}, …, x_{m}^{1/r}), r = n!.

But, by Proposition 1, the roots of the restriction of f(x; t) to each polynomial arc γ through zero can be arranged as n Q-analytic functions at zero. Hence, the restrictions of the fractional power series Ψ_{i}(x^{1/r}) to each polynomial arc γ through zero are real formal power series. Therefore Ψ_{1}(x), …, Ψ_{n}(x) ∈ ℝ[[x]] by virtue of the lemma below.

Lemma 2. Let

Ψ(x, y) = ∑_{i,α} a_{i,α}x^{i/r}y^{α/r} ∈ ℂ[[x^{1/r}, y^{1/r}]], i ∈ ℕ, α ∈ ℕ^{m}, r ∈ ℕ, r > 1,

be a fractionary power series, where x = x_{1} and y = (y_{1}, …, y_{m}). Suppose that a_{j,β} ≠ 0 for some j ∈ ℕ \ rℕ, β ∈ ℕ^{m}. Then there is a polynomial arc γ such that Ψ ∘ γ is fractionary but not a power series.

Indeed, consider polynomial arcs γ parametrized by

x = τ, y = (cτ)^{r}, c ∈ ℂ^{m}.

Then

(Ψ ∘ γ)(τ) = ∑_{p} ( ∑_{i/r+|α|=p} a_{i,α}c^{α}) τ^{p}.

Put q := j/r + |β|. Clearly, the polynomial

P (C) := ∑_{i/r+|α|=q} a_{i,α}C^{α} ≢ 0, C = (C_{1}, …, C_{m}),

does not vanish, because a_{j,β} ≠ 0, and if i/r + |α| = j/r + |β| and α = β, then i = j. Now it suffices to take c ∈ ℂ^{m} for which P (c) ≠ 0.

The ring ℝ[[τ]] of formal power series in one variable τ is an ordered ring by putting τ > 0. Consider formal arcs γ(τ) = (γ_{1}(τ), …, γ_{m}(τ)) ∈ (ℝ[[τ]])^{m} through zero. In the following lemma about the roots Ψ_{i}(x), the open cubes Q_{j} are regarded, by abuse of notation, as the subsets of (ℝ[[τ]])^{m} given by the same inequalities as those of ℝ^{m}.

Lemma 3. Suppose a formal arc γ(τ) through zero lies in an open cube Q_{j}. If Ψ_{i}(γ) < Ψ_{k}(γ), then this inequality remains true for every formal arc ϑ(τ) through zero lying in that cube. In other words, one can order the formal roots Ψ_{i}(x), i = 1, …, n, with respect to any open cube Q_{j}.

We first observe that the formal power series Ψ_{i}(x) can be approximated in the Krull topology by convergent power series Ψ_{i,ν}(x), i = 1, …, n, ν ∈ ℕ, such that the discriminants of the polynomials

f_{ν}(x) := ∏_{i=1}^{n}(t − Ψ_{i,ν}(x)), ν ∈ ℕ,

are of the form D_{ν}(x) = x^{α} · u_{ν}(x) with u_{ν}(0) ≠ 0. Indeed, the discriminant D(x) = x^{α} · u(x) with u(0) ≠ 0, of the polynomial f(x, t) can be expressed as follows:

D(x) = P (Ψ_{1}(x), …, Ψ_{n}(x)) ∈ ℝ[[x]],

where P (T_{1}, …, T_{n}) ∈ ℤ[T_{1}, …, T_{n}] is a polynomial with integer coefficients; here we identify Q-analytic function germs at zero with their Taylor series. Our assertion thus follows from the Artin approximation theorem applied to the system of two polynomial equations

P (A_{1}, …, A_{n}) = x^{α} · U, U · V = 1,

with indeterminates A_{1}, …, A_{n}, U, V.

Therefore, Ψ_{i,ν}(γ) < Ψ_{k,ν}(γ) for ν large enough. By approximation in the Krull topology, we may assume that γ is a convergent arc through zero. It is clear that this inequality remains true for every convergent arc through zero lying in the cube Q_{j}. Let ϑ be a formal arc through zero lying in Q_{j}. Then, again by approximation in the Krull topology, Ψ_{i,ν}(ϑ) ≤ Ψ_{k,ν}(ϑ), and hence Ψ_{i}(ϑ) ≤ Ψ_{k}(ϑ). But Ψ_{i}(ϑ) ≠ Ψ_{k}(ϑ) because the discriminant D ∘ ϑ ≠ 0 of the restriction of the polynomial f(x; t) to the formal arc ϑ is not zero. Lemma 3 thus follows.

Lemma 3 is valid, of course, not only at 0 ∈ U, but at each point a ∈ U. Theorem 1 will be established once we prove that the arc-symmetric Q-set X has, in the AS topology, precisely n irreducible components C_{i}, i = 1, …, n, each of which is the graph of an arc-Q-function.

To this end, observe that in the desingularization from Theorem 2 one can ensure that the set

X̃ \ ⋃_{ji} σ^{−1}(graph (ψ_{j,i}))

is a normal crossing divisor (locally, the zero locus of the Jacobian determinant of σ or a submanifold of codimension 1; cf. [5], Theorem 5.10). Since σ is a proper modification which is a Q-diffeomorphism over each graph (ψ_{j,i}), we get

σ(∂(σ^{−1}(graph (ψ_{j,i}))) = ∂(graph (ψ_{j,i}));

here ∂A denotes the frontier of a set A. Let Q_{j}, Q_{l} be two adjacent cubes with common face Q_{j,l}. We first consider our problem of irreducible components in the arc-symmetric topology over the open subset Ω_{j,l} := Q_{j} ∪ Q_{l} ∪ Q_{j,l}. In a similar fashion, we can proceed with attaching successive cubes.

We introduce the following notation:

X_{j} := X ∩ (Q_{j} × ℝ), X_{l} := X ∩ (Q_{l} × ℝ), X_{j,l} := X ∩ (Ω_{j,l} × ℝ)

X̃_{j} := X̃ ∩ σ^{−1}(Q_{j} × ℝ), X̃_{l} := X̃ ∩ σ^{−1}(Q_{l} × ℝ), X̃_{j,l} := X̃ ∩ σ^{−1}(Ω_{j,l} × ℝ),

A_{j,i} = graph (ψ_{j,i}) and Ã_{j,i} = σ^{−1}(A_{j,i}).

It is not difficult to check that the closure (in the ordinary topology) of each set A_{j,i} is the graph of a continuous function on Q_{j} ∪ Q_{j,l}. Since the roots of a monic polynomial depend continuously on the coefficients, we get ∂X_{j} = ∂X_{l}; here frontier ∂ is understood with respect to the ambient space Ω_{j,l} × ℝ. Then

X̃_{j} = ⋃_{i} Ã_{j,i}, X̃_{l} = ⋃_{k} Ã_{l,k}

and X̃_{j,l} \ (X̃_{j} ∪ X̃_{l}) is a normal crossing divisor. Clearly,

∂Ã_{j,i} = ⋃_{k≠i}(∂Ã_{j,i} ∩ ∂Ã_{j,k}) ∪ ⋃_{k}(∂Ã_{j,i} ∩ ∂Ã_{l,k}),

whence

∂A_{j,i} = σ(∂Ã_{j,i}) = ⋃_{k≠i} σ(∂Ã_{j,i} ∩ ∂Ã_{j,k}) ∪ ⋃_{k} σ(∂Ã_{j,i} ∩ ∂Ã_{l,k}).

We still need the following

Claim. The sets ∂Ã_{j,i}, i = 1, …, n, are pairwise disjoint, and the sets σ(∂Ã_{j,i} ∩ ∂Ã_{l,k}), k = 1, …, n, are pairwise disjoint too.

Suppose, on the contrary, that two sets ∂Ã_{j,i} and ∂Ã_{j,p}, with p ≠ i, have a common point P̃. Take a Q-analytic arc η : (−1, 1) ⟶ X̃_{j,l} such that η(−1, 0) ⊂ Ã_{j,p} , η(0, 1) ⊂ Ã_{j,i} and η(0) = P̃. Then

(σ ∘ η)(τ) = (γ(τ), ψ(τ)), τ ∈ (−1, 1),

for some Q-analytic γ and ψ. Let π : ℝ_{x}^{m} × ℝ_{t} ⟶ ℝ_{x}^{m} be the canonical projection.

We may assume, by a linear change of variables, that (π∘σ)(P̃) = 0 ∈ ℝ^{m}. Keep the notation of Lemma 3 and order the formal roots Ψ_{i}(x), i = 1, …, n, with respect to the open cube Q_{j}. The arcs γ(τ) and γ(−τ) lie in the cube Q_{j}, ψ(τ) occurs at the i-th place among Ψ_{1}(γ(τ)), …, Ψ_{n}(γ(τ)), and ψ(−τ) occurs at the p-th place among Ψ_{1}(γ(−τ)), …, Ψ_{n}(γ(−τ)), which contradicts the conclusion of Lemma 3.

To prove the second assertion, suppose that two sets

σ(∂Ã_{j,i} ∩ ∂Ã_{l,k}) and σ(∂Ã_{j,i} ∩ ∂Ã_{l,q}) for some q ≠ k,

have a common point P = σ(P̃) = σ(S̃) with

P̃ ∈ ∂Ã_{j,i} ∩ ∂Ã_{l,k} and S̃ ∈ ∂Ã_{j,i} ∩ ∂Ã_{l,q}.

Take two Q-analytic arcs η, ζ : (−1, 1) ⟶ X̃_{j,l} such that η(−1, 0) ⊂ Ã_{l,k}, η(0, 1) ⊂ Ã_{j,i}, η(0) = P̃, ζ(−1, 0) ⊂ ∂Ã_{l,q}, ζ(0, 1) ⊂ ∂Ã_{j,i} and ζ(0) = S̃. Then

(σ ∘ η)(τ) = (γ(τ), ψ(τ)), τ ∈ (−1, 1)

and

(σ ∘ ζ)(τ) = (ϑ(τ), φ(τ)), τ ∈ (−1, 1),

for some Q-analytic γ, ϑ, ψ and φ.

As before, assume that π(P) = 0 ∈ ℝ^{m}. The arcs γ(τ), ϑ(τ) lie in the cube Q_{j}, and the arcs γ(−τ), ϑ(−τ) lie in the cube Q_{l}. Clearly,

ψ(τ) = Ψ_{i}(γ(τ)) and φ(τ) = Ψ_{i}(ϑ(τ)),

and thus

ψ(−τ) = Ψ_{i}(γ(−τ)) and φ(−τ) = Ψ_{i}(ϑ(−τ)).

Hence and by Lemma 3, the roots ψ(−τ) and φ(−τ) for τ > 0 belong to a common set A_{l,p}. This contradiction completes the proof of the claim.

In view of the above claim, we see that the connected set ∂A_{j,i} is the disjoint union of the closed subsets σ(∂Ã_{j,i} ∩∂Ã_{l,k}), k = 1, …, n. Consequently,

∂A_{j,i} = σ(∂Ã_{j,i} ∩ ∂Ã_{l,p})

for a unique p = ω(i) ∈ {1, …, n}, and

∂Ã_{j,i} ∩ ∂Ã_{l,k} = ∅ for all k = 1, …, n, k ≠ ω(i).

Hence ∂Ã_{j,i} ⊂ ∂Ã_{l,ω(i)}, and thus, by symmetry, we get ∂Ã_{j,i} = ∂Ã_{l,ω(i)}. Consequently,

Ã_{j,i} ∪ Ã_{l,ω(i)} ∪ ∂Ã_{j,i}

is the connected component (in the ordinary topology) of X̃_{j,l}. Hence and by Theorem 2,

A_{j,i} ∪ A_{l,ω(i)} ∩ ∂A_{j,i}

is the irreducible component (in the arc-symmetric topology) of X_{j,l} containing A_{j,i}. Therefore the arc-symmetric Q-subset X_{j,l} of Ω_{j,l} × ℝ has precisely n irreducible arc-symmetric components C_{i}, i = 1, …, n, of the form

C_{i} = A_{j,i} ∪ A_{l,ω(i)} ∩ ∂A_{j,i} = graph (χ_{i}),

where i ↦ ω(i) is a permutation of {1, …, n} and each χ_{i} is a continuous function which extends the functions ψ_{j,i} and ψ_{l,ω(i)}.

It remains to show that each function χ_{i} is an arc-Q-function with the property (T). So let γ : (−1, 1) ⟶ Ω_{j,l} be a Q-analytic arc and put:

X_{γ} := {(τ, t) ∈ (−1, 1) × ℝ : f(γ(τ), t) = 0} = (γ × Id)^{−1}(X).

By Proposition 1, X_{γ} is the union of the graphs of n, not necessarily distinct, Q-analytic functions on (−1, 1), say, φ_{1}(τ), …, φ_{n}(τ). Then

C_{i} ⊂ ⋃_{k=1}^{n}(γ, φ_{k})(−1, 1) and ⋃_{k=1}^{n}(γ, φ_{k})^{−1}(C_{i}) = (−1, 1).

Hence Int (γ, φ_{k})^{−1}(C_{i}) ≠ ∅ for some k, and thus (γ, φ_{k})(−1, 1) ⊂ C_{i}, because C_{i} is an arc-symmetric Q-subset. This demonstrates that

(χ_{i} ∘ γ)(τ) = φ_{k}(τ) for all τ ∈ (−1, 1),

and thus χ_{i} ∘ γ is a Q-analytic function, as required. Property (T) follows directly from Lemma 3.

Eventually, we can repeat the above arguments when attaching successive cubes Q_{p}. In this manner, we see that X has precisely n irreducible components in the AS topology, each of which is the graph of an arc-Q-function. This completes the proof of Theorem 1.

Now, let A be a normal domain, K its quotient field and f(t) ∈ A[t] be a monic polynomial. It is well known from commutative algebra that if f(t) is irreducible over A, then so it is over K. Consequently, if the field K is of characteristic zero, then every irreducible monic polynomial f(t) ∈ A[t] is square-free.

We cannot apply this algebraic assertion directly to the domain Q(Ω). However, each Q-meromorphic fraction

a(x)/b(x), a(x), b(x) ∈ Q(Ω),

which is integral over Q(Ω), can be transformed to a Q-analytic function over any open, relatively compact subset U ⊂ Ω by a finite sequence of global blowings-up with smooth centers. Indeed, one can transform the functions a(x) and b(x) by global blowing up to normal crossings, which are locally of the form y^{α} · u(y) and y^{β} · v(y), respectively, where u(y), v(y) are units and either α ≤ β or α ≥ β. In our case, since the fraction y^{α−β} · u(y)v^{−1}(y) is integral over the ring of Q-analytic functions, there must be α ≥ β, which is the desired result. Consequently, one can adapt the foregoing algebraic assertion to the quasianalytic settings as follows.

Proposition 3. (On square-free factorization) If f(x; t) is a monic polynomial with coefficients in Q(Ω) and U ⊂ Ω is an open, relatively compact subset, then there exists a Q-modification σ : Ũ ⟶ U such that

i) σ is a composite of finitely many global blowings-up with smooth centers; ii) the pull-back polynomial f^{σ}(y; t) factorizes globally into a product of

square-free monic polynomials with coefficients in Q(Ũ).

Hence and via transformation to normal crossings by blowing up, we obtain the following

Corollary 3. Every monic polynomial f(x; t) ∈ Q(Ω)[t] with quasianalytic coefficients factorizes, after a suitable transformation of its coefficients by a finite sequence of global blowings-up with smooth centers over any open, relatively compact subset U ⊂ Ω, into a product of quasiordinary polynomials with quasianalytic coefficients. Moreover, if the polynomial f(x; t) is hyperbolic, so are its modified quasiordinary factors.

Corollary 3 and Theorem 1 yield immediately the following theorem on the splitting of hyperbolic polynomials:

Corollary 4. If f(x; t) is a hyperbolic polynomial with coefficients in Q(Ω) and U ⊂ Ω is an open, relatively compact subset, then there exists a Q-modification σ : Ũ ⟶ U such that

i) σ is a composite of finitely many blowings-up with smooth centers; ii) the pull-back polynomial f^{σ}(y; t) has, locally in the vicinity of each

point b ∈ Ũ, a factorization of the form

f(y; t) = ∏_{i=1}^{n}(t − ψ_{i}(y)),

where ψ_{i}(y) are arc-Q-functions near b.

Before drawing subsequent corollaries, we state two theorems on rectilinearization of quasianalytic functions and arc-quasianalytic functions. The latter will play a key role in our approach to hyperbolic polynomials. They are counterpart of two theorems by Bierstone–Milman [3] from the real analytic geometry, and are proven in a manuscript by Bierstone–Milman– Valette [7] yet unpublished. It should be emphasized, however, that the quasianalytic versions are much weaker than the analytic ones, because they refer only to quasianalytic functions which satisfy a non-trivial Q-analytic equation. The passage from functions with subanalytic graphs to functions with semianalytic graphs needs much stronger algebro-analytic methods unavailable in the quasianalytic settings such as, for instance, complexification and the techniques of flattening or equidimensionality (see also [19, 20]).

Theorem 3. (Rectilinearization of Q-subanalytic functions) Let M be a (connected) Q-manifold, U ⊂ M a relatively compact, open, definable subset, and g : M ⟶ ℝ be a continuous Q-subanalytic function which satisfies a Q-analytic equation of the form Φ(x, g(x)) = 0, Φ ≢ 0. Then there exists a surjective Q-mapping σ : Ũ ⟶ U such that

i) σ is a composite of finitely many Q-mappings, each of which is either a blowing-up with smooth center or a surjection of the form ∐ U_{j} ⟶ ⋃ U_{j}, where (U_{j})_{j} is a finite covering of the target space by coordinate charts and ∐ denotes disjoint union, or a local power substitution of the form

U_{j} ⟶ U_{j}, (u_{1}, …, u_{m}) ↦ (ε_{1}u_{1}^{r_{1}}, …, ε_{m}u_{m}^{r_{m}}),

where (u_{1}, …, u_{m}) are the coordinates of U_{j}, r_{1}, …, r_{m} > 0 are integers and ε_{1} …, ε_{m} = ±1;

ii) g ∘ σ is a Q-analytic function.

Theorem 4. (Description of arc-Q-analytic functions) Let M be a (connected) Q-manifold, U ⊂ M a relatively compact, open, definable subset, and g : M ⟶ ℝ be an arc-Q-analytic function which satisfies a Q-analytic equation of the form Φ(x, g(x)) = 0, Φ ≢ 0. Then there exists a surjective Q-mapping σ : Ũ ⟶ U such that

i) σ is a composite of finitely many Q-mappings, each of which is either a blowing-up with smooth center or a surjection of the form ∐ U_{j} ⟶ ⋃ U_{j}, where (U_{j})_{j} is a finite covering of the target space by coordinate charts and ∐ denotes disjoint union;

ii) g ∘ σ is a Q-analytic function.

From Corollary 4 and Theorem 4 we can immediately derive the following two results, which carry over those from [12] to the quasianalytic settings.

Corollary 5. If f(x; t) is a hyperbolic polynomial with coefficients in Q(Ω) and U ⊂ Ω is an open, relatively compact subset, then there exists a Q-modification σ : Ũ ⟶ U such that

i) σ is a composite of finitely many Q-mappings, each of which is either a blowing-up with smooth center or a surjection of the form ∐ U_{j} ⟶ ⋃ U_{j}, where (U_{j})_{j} is a locally finite covering of the target space by coordinate charts and ∐ denotes disjoint union;

ii) the pull-back polynomial f^{σ}(y; t) has locally, in the vicinity of each point b ∈ Ũ, a Q-analytic factorization of the form

f^{σ}(y; t) = ∏_{i=1}^{n}(t − ψ_{i}(y)).

Corollary 6. Under the foregoing assumptions, there exists a Qsubanalytic subset Σ ⊂ Ω of codimension at least 2 such that, in the vicinity of each point a ∈ Ω \ Σ, the hyperbolic polynomial f(x; t) has a Q-analytic factorization of the form

f(x; t) = ∏_{i=1}^{n}(t − ψ_{i}(x)).

We shall now apply the foregoing description of arc-Q-analytic functions and the quasianalytic version of Glaeser’s composite function theorem presented in our paper [21] to achieve the following strengthening of Theorem 1.

Theorem 1^{∗}. Let Ω be an open, simply connected subset of ℝ^{m}. Then every hyperbolic quasiordinary polynomial

f(x; t) = t^{n} + a_{n−1}(x)t^{n−1} + ⋯ + a_{0}(x) ∈ Q(Ω)[t]

splits into linear factors of the form

f(x; t) = ∏_{i=1}^{n}(t − ψ_{i}(x)), x ∈ Ω,

where ψ_{i}(x) are smooth (i.e. of class C^{∞}) functions on Ω definable in the structure R_{Q}.

As in the case of Theorem 1, we may confine our proof to an open, relatively compact subset

U = (−δ, δ)^{m} × (−M, M)

of Ω×ℝ which contains all roots of f(x; t) over (−δ, δ)^{m}. Consider a surjective Q-mapping σ : Ũ ⟶ U which is a finite composite of local blowings-up along smooth centers such that the superpositions

ψ̃_{i} := ψ_{i}(x) ∘ σ : Ũ ⟶ ℝ, i = 1, …, n,

are Q-analytic functions. We show that each function ψ̃_{i} is formally composite with σ, i.e. for each P ∈ U, there is a formal power series Φ at P such that, for every point P̃ ∈ σ^{−1}(P) on the fiber over P, the Taylor series TP̃ ψ̃_{i} of ψ̃_{i} at P̃ coincides with Φ ∘ TP̃ σ. As Φ above, we should take the formal power series Ψ_{i} from the conclusion of Theorem 1, property (T). Indeed, for a generic straight line l(τ) through P̃, l(0) = P̃, we have

Ψ_{i} ∘ TP̃ σ ∘ l = Ψ_{i} ∘ TP̃ (σ ∘ l) = TP̃ (ψ_{i} ∘ σ ∘ l) = TP̃ (ψ̃_{i} ∘ l) = TP̃ ψ̃_{i} ∘ l.

Hence Ψ_{i} ∘ TP̃ σ = TP̃ (ψ̃_{i}), as required.

Since the mapping σ is generically a submersion, it follows from the quasianalytic version of the composite function theorem from [21] that ψ̃_{i} is a composite function with σ, i.e. there is a smooth mapping

φ_{i} : U ⟶ ℝ such that φ̃_{i} = φ_{i} ∘ σ.

Therefore, the mappings φ_{i} and ψ_{i} coincide over an open and dense subset of U, whence so do they over U. This completes the proof.

Before turning to the perturbation of symmetric matrices, we introduce some terminology. We shall state the results about perturbation of matrices with quasianalytic entries in terms of quasianalytic function germs. Denote by M_{m,a} the field of quasi-meromorphic germs at a ∈ ℝ^{m}, i.e. the quotient field of the local ring Q_{m,a} of Q-analytic germs at a point a ∈ ℝ^{m}. In the vector spaces ℝ^{n} and (M_{m,a})^{n} over the fields ℝ and M_{m,a}, respectively, the standard inner products given by the formula

v • w = v_{1}w_{1} + ⋯ + v_{n}w_{n} and f(x) • g(x) = f_{1}(x)g_{1}(x) + ⋯ + f_{n}(x)g_{n}(x).

The spectral theorem for symmetric matrices is valid for any real closed field. The assumption of real closedness is necessary to ensure that the characteristic polynomial of a given symmetric matrix, which always is hyperbolic, factorizes into linear factors. We are able to dispense with it, but we must apply Corollary 5 instead, i.e. split the characteristic polynomial into linear factors via transformation by blowing up. Therefore, repeating mutatis mutandis the proof of the spectral theorem from linear algebra, we obtain the following counterpart over the field of quasi-meromorphic function germs.

Theorem 5. (Spectral theorem with quasianalytic parameters.) Let N be a symmetric n × n matrix with quasianalytic entries from Q_{m,0}. Then we can find a modification σ : W ⟶ U of a neighbourhood U of 0 ∈ ℝ^{m}, which is a finite composite of local blowings-up with smooth centers, such that for each point b ∈ σ^{−1}(0) ⊂ W the vector space (M_{W,b})^{n} over the field M_{W,b} has an orthogonal basis

w^{1}(y), …, w^{n}(y) ∈ (Q_{W,b})^{n}

that consists of eigenvectors of the pull-back matrix N^{σ}.

A matrix with entries from the ℝ-algebra Q_{m,a} may be regarded as a quasianalytic family of real matrices parametrized by x. Our next objective is to achieve a simultaneous quasianalytic diagonalization of the pull-back matrix N^{σ} after performing a suitable modification σ which is a finite composite of local blowings-up with smooth centers. Let

λ_{1}(y), …, λ_{n}(y) ∈ Q_{W,b}

be the eigenvalues of N^{σ}, which may not be pairwise distinct. The above theorem yields a quasianalytic family

w^{1}(y), …, w^{n}(y) ∈ ℝ^{n}

of orthogonal eigenvectors which form a basis of ℝ^{n} generically near b ∈ W; say over a set W_{0} = W \ Σ where Σ ⊂ W is a closed Q-analytic subset of codimension at least one. Fix a vector w(b) = w^{j}(b), j = 1, …, n, and its eigenvalue λ(b) = λ_{j}(b). Take any sequence (b_{k}) ⊂ W_{0} which tends to b: b_{k} → b, and such that the limit

v(b) = lim_{k→∞} v(b_{k}), where v(b_{k}) = \frac{w(b_{k})}{∥w(b_{k})∥} ,

exists. We obviously have

(N^{σ}(b) − λ(b)) · v(b) = lim_{k→∞}(N^{σ}(b_{k}) − λ(b_{k})) · v(b_{k}) = 0.

Since the vectors w^{j}(y) are pairwise orthogonal, we obtain in this manner an orthonormal basis

v^{1}(b), …, v^{n}(b) ∈ ℝ^{n}

that consists of eigenvectors of the matrix N^{σ}(b).

We wish to construct a quasianalytic family

v^{1}(y), …, v^{n}(y) ∈ ℝ^{n}

of orthonormal bases that consist of eigenvectors of the matrices N^{σ}(y). Clearly, this will be possible once we know that all the components of each vector w^{j}(y), j = 1, …, n, are divisible in Q_{W,b} by one of them. It is well known that the last condition can be ensured via a successive transformation to normal crossings by blowing up. Consequently, we have obtained the theorem stated below, which generalizes to the quasianalytic settings the result of Kurdyka–Paunescu [12] about real analytic perturbation of symmetric matrices.

Theorem 6. (On quasianalytic diagonalization of symmetric matrices) Consider a symmetric n × n matrix N with entries from Q_{m,0}. Then there exists a modification σ : W ⟶ U of a neighbourhood U of zero, which is a finite composite of local blowings-up with smooth centers, such that the pullback matrix N^{σ} admits a simultaneous quasianalytic diagonalization near each point b ∈ σ^{−1}(0) ⊂ W. This diagonalization can be performed by a Q-analytic choice of orthonormal bases that consist of eigenvectors of N^{σ}.

Remark 2. Both Theorem 5 and 6 with quasianalytic parameters remain valid, with the same proof, in the case of quasianalytic families of hermitian matrices with entries from Q_{m,0} ⊗_{ℝ} ℂ.

Remark 3. All the above results can be, as shown by Kurdyka–Paunescu, carried over to the case of polynomials with purely imaginary roots, and thence to that of antisymmetric matrices. Indeed, a polynomial

f(x; t) = t^{n} + a_{n−1}(x)t^{n−1} + ⋯ + a_{0}(x) ∈ Q_{m,0}[t]

has purely imaginary roots iff the polynomial i^{−n}f(x; it) is hyperbolic (cf. [12] for details).

Remark 4. It is well known that, in general, one cannot find bases of eigenvectors even in a continuous way. This is caused by that the angle between linearly independent eigenvectors, which correspond to distinct eigenvalues, may tend to zero when approaching a given point b.

Acknowledgements. The author wishes to express his gratitude to the referee for several valuable suggestions. This research was partially supported by Research Project No. N N201 372336 from the Polish Ministry of Science and Higher Education.

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Institute of Mathematics Jagiellonian University ul. Profesora Łojasiewicza 6 30-348 Kraków, Poland e-mail address: nowak@im.uj.edu.pl