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

290§9. DUAL SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS9.1. Solvability with respect to the variables. Let M be as in §2.10 definedby y = Π(x, a, b) or dually by b = Π ∗ (a, x, y).Definition 9.2. M is solvable with respect to the variables if there exist aninteger κ ∗ 1 and multiindices δ(1), . . .,δ(n) ∈ N p with |δ(l)| 1 for l =1, . . ., n and max 1ln |δ(l)| = κ ∗ , together with integers j(1), . . .,j(n)with 1 j(l) m such that the local K-analytic map(9.3) ( (ΠK n+m ∋ (x, y) ↦−→ ∗j (0, x, y) ) ( ) )1jm, Π ∗ j(l)(0, x, y) ∈ K m+na δ(l) 1lnis of rank equal to n + m at (x, y) = (0, 0)If M is a complexified generic submanifold, solvability with respect tothe parameters is equivalent to solvability with respect to the variables, becauseΠ ∗ = Π. This is untrue in general: with n = 2, m = 1, consider thesystem y x 2 = 0, y x 1 x 1 = 0, whose general solutions is y(x) = b + x 1a withx 2 absent.To characterize generally such a degeneration property, we develope both(9.4)y j = Π j (x, a, b) = ∑ x β Π j β(a, b) andβ∈N nb j = Π ∗j (a, x, y) = ∑ a δ Π ∗j δ(x, y),δ∈N pwith analytic functions Π j β (a, b), Π∗ jδ (x, y) and we introduce two K∞ -valuedmaps(9.5)Q ∞ :Q ∗ ∞ :(a, b) ↦−→ ( Π j β (a, b)) 1jmβ∈N n(x, y) ↦−→ (Π ∗j δ (x, y)) 1jmδ∈N p .Since b ↦→ ( Π j 0(0, b) ) 1jmand y ↦→(Π∗j0(0, y) ) 1jmare already both ofrank m at the origin, the generic ranks of these two maps, defined by testingthe nonvanishing of minors of their infinite Jacobian matrices, satisfy(9.6)genrkQ ∞ = m + p MgenrkQ ∗ ∞ = m + n Mfor some two integers 0 p M p and 0 n M n. So at a Zariski-genericpoint, the ranks are equal to m + p M and to m + n M .Proposition 9.7. There exists a local proper K-analytic subset Σ M of K n x ×K m y × K p a × K m b whose equations, of the specific form{}(9.8) Σ M = r ν (a, b) = 0, ν ∈ N, rµ(x, ∗ y) = 0 µ ∈ N ,andand