Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

11.4. Regularity and jet parametrization. Some strong rigidity propertiesunderly ( ) the above ( diagram. ) Especially, the smoothness of the two pairsFv , F p and F′v , F ′ p governs the smoothness of (ϕ, h).(We shall)study the regularity of a purely formal map (z ′ , c ′ ) =ϕ(z), h(c) , namely ϕ(z) ∈ K[z] n+m and h(c) ∈ K[c] p+m , assuming (E )and (E ′ ) to be analytic. Concretely, the assumption that (ϕ, h) maps M toM ′ reads as one of the four equivalent identities:⎧ψ ( x, Π(x, c) ) ≡ Π ′( φ(x, Π(x, c)), h(c) ) ,⎪⎨ ψ(z) ≡ Π ′( φ(z), h(a, Π ∗ (a, z) ) ,(11.5)g ( a, Π ∗ (a, z) ) ≡ Π ′ ∗ ( f(a, Π ∗ (a, z)), ϕ(z) ) ,⎪⎩g(c) ≡ Π ′ ∗ ( f(c), ϕ(x, Π(x, c)) ) ,in K[x, c] m and in K[a, z] m .Theorem 11.6. Let (ϕ, h) := M → M ′ be a purely formal equivalencebetween two local K-analytic submanifolds of solutions. Assume that thefundamental pair of foliations (F v , F p ) is covering at the origin, with type(µ, µ ∗ ) at the origin. Assume that M ′ is both κ-solvable with respect to theparameters and κ ∗ -solvable with respect to the variables. Set l := µ ∗ (κ +κ ∗ ) and l ∗ := µ(κ ∗ + κ). Then there exist two K n+m -valued and K p+m -valued local K-analytic maps Φ l and H l ∗, constructible only by means ofΠ, Π ∗ , Π ′ , Π ′∗ , such that the following two formal power series identitieshold:{ (ϕ(z) ≡ Φl z, Jlz ϕ(0) ) ,(11.7)(h(c) ≡ H l ∗ c, Jl ∗c h(0) ) ,in K[z] n+m and in K[c] p+m , where Jz l ϕ(0) denotes the l-th jet of h at theorigin and similarly for Jc l∗ h(0). In particular, as a corollary, we have thefollowing two automatic regularity properties:• ϕ(z) ∈ K{z} n+m and h(c) ∈ K{c} p+m are in fact convergent;• if in addition M and M ′ are K-algebraic in the sense of Nash, thenΦ l and H l ∗ are also K-algebraic, whence ϕ(z) ∈ A K {z} n+m andh(c) ∈ A K {c} p+m are in fact K-algebraic.Proof. We remind the explicit expressions of the two collections of vectorfields spanning the leaves of the two foliations F v and F p :⎧L k := ∂ m∑ ∂Π j+ (x, c) ∂⎪⎨ ∂x k ∂x k ∂y , k = 1, . . .,n,j(11.8)⎪⎩j=1L ∗ q := ∂∂a q + m∑j=1∂Π ∗j ∂(a, z)∂aq ∂b , jq = 1, . . .,p.297