2 3 Fe b 20 06 HYPERBOLIC POLYNOMIALS AND MULTIPARAMETER REAL ANALYTIC PERTURBATION THEORY

Let P (x, z) = z + ∑d i=1 ai(x)z d−i be a polynomial, where ai are real analytic functions in an open subset U of R. If for any x ∈ U the polynomial z 7→ P (x, z) has only real roots, then we can write those roots as locally lipschitz functions of x. Moreover, there exists a modification (a locally finite composition of blowing-ups with smooth centers) σ : W → U such that the roots of the corresponding polynomial P̃ (w, z) = P (σ(w), z), w ∈ W , can be written locally as analytic functions of w. Let A(x), x ∈ U be an analytic family of symmetric matrices, where U is open in R. Then there exists a modification σ : W → U , such the corresponding family Ã(w) = A(σ(w)) can be locally diagonalized analytically (i.e. we can choose locally a basis of eigenvectors in an analytic way). This generalizes the Rellich’s well known theorem (1937) for one parameter families. Similarly for an analytic family A(x), x ∈ U of antisymmetric matrices there exits a modification σ such that we can find locally a basis of proper subspaces in an analytic way.