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

80yxx j of the form(5.5) {m∑0 = A j +l 1 =1y l 1xx ·[−X y l 1 Y jx + Y jy l 1 X x +m∑l 2 =1[]]y l 2x · −X y l 1 Y j y l 2 y l 1 y l 2 , ,for j = 1, . . .,m, where A j is an abbreviation for the terms appearing inthe lines 5, 6, 7, 8, 9 and 10 of (5.3), or even more compactly, changing theindex j to the index k{(5.6)0 = A k +m∑l 1 =1y l 1xx · B k l 1,for k = 1, . . .,m, where Bl k 1is an abbreviation for the terms in the bracketsin (5.5).Thanks to the assumption that the determinant (3.2) is the identity determinantat (x, y) = (0, 0), we deduce that the determinant of the m × mmatrix (Bl k 1) 1km1l 1 m is also the identity determinant at (x, y, y x) = (0, 0, 0). Itfollows that the determinant of the m × m matrix (Bl k 1) 1km1l 1 mis nonvanishingin a neighborhood of the origin in the first order jet space. Consequently,we can apply the rule of Cramer to solve the yxx j explicitely interms of theA k and of the Bl k 1as follows⎧ ∣ ∣∣∣∣∣ B1 1 · · · A 1 · · · B 1 m· · · · · · · · · · · · · · ·⎪⎨(5.7) yxx j = − B1 m · · · A m · · · Bmm ∣B1 1 · · · B 1 · · · B 1 m⎪⎩· · · · · · · · · · · · · · ·∣ B1 m · · · B m · · · Bmm ∣where on the numerator, the only modification of the determinant of the matrix(Bl k 1) 1km1l 1 mis the replacement of its j-th column by the column vectorA. We have to show that after replacing the A k and the Bl k 1by their completeexpressions, one indeed obtains the desired equation (5.4). As in (3.43), weshall introduce a notation for the two m×m determinants appearing in (5.6):we write this quotient under the form{ ∣ ∣B k(5.8) yxx j = − 1 | · · · | j A k | · · · |Bmk ∣ ∣ ∣ Bk1 | · · · | j Bj k| · · · |Bk ∣ ∣ ,mwhere it is understood that B k 1, . . .,B k j , . . ., B k m and A k are column vectorswhose index k (for their lines) varies from 1 to m. This representation ofdeterminants emphasizing only its columns will be appropriate for later manipulations.