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

As explained before the statement of Theorem 4.19, the subscript α isobtained as follows. The term σ(d : µ d : λ d ) is of the form (e : ν d : γ e ), forsome e with 1 e d, for some ν e with 1 ν e µ e and for some γ e with1 γ e λ e . The single pure jet variable351(5.7) y le:νek e:νe:1 ,...,k e:νe:γe ,...,k e:νe:λeappears inside the total monomial(5.8)∏y l 1:ν 1k 1:ν1 :1,...,k 1:ν1· · ·:λ 11ν 1 µ 1∏y l d:ν dk d:νd :1,...,k d:νd,:λ d1ν d µ dplaced at the end of the formula for Y j i 1 ,...,i κ(see in advance formula (5.13)below; this total monomial generalizes to several dependent variables thetotal monomial appearing in the last line of (3.74)). According to the ruleexplained before the statement of Theorem 4.18, the index l α must be equalto l e:νe , since l e:νe is attached to the monomial (5.7). Coming back to theterm σ(d:µ d :λ d ), we shall denote this index by(5.9) l e:νe =: l π(e:νe:γ e) =: l πσ(d:µd :λ d ),where the symbol π denotes the projection from the set(5.10) {1:1:1, . . ., 1:µ 1 :λ 1 , . . .. . .,d:1:1, . . .,d:µ d :λ d }to the set(5.11) {1:1, . . ., 1:µ 1 , . . .,d:1, . . ., d:µ d }simply defined by π(e:ν e :γ e ) := (e:ν e ).In conclusion, by means of this formalism, we may write down the completegeneralization of Theorems 2.24, 3.73 and 4.18 to several independentvariables and several dependent variablesTheorem 5.12. For j = 1, . . ., m, for every κ 1 and for every choice of κindices i 1 , . . .,i κ in the set {1, 2, . . ., n}, the general expression of Y j i 1 ,...,i κ