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

Replacing this expression of A k in (5.8), taking account of the expression ofthe denominator already obtained in the second line of (5.11) and abbreviating∆(x|y 1 | · · · |y m ) as ∆, we may write (5.8) in length and then develope itby linearity as follows(5.15)⎧⎪⎨⎪⎩y j xx ==−1[DX] m−1 · ∆ ·−1[DX] m−1 · ∆ ·⎡⎢⎣⎡⎢⎣∣ ∣ ∣ Yky 1 · DX − X y 1 · DY k | · · ·· · · | j DX · Yxx k − DY k · Y km∑+2+l 1 =1m∑y l 1x ·m∑l 1 =1 l 2 =1∣ ∣ ∣ Yky 1 · DX − X y 1 · DY k | · · ·+ 2+xx+83[DX · Y kxy l 1− DY k · X xy l 1]+⎥· · · |Yy k · DX − X m y m · DY k∣ ∣ ∣ ⎦⎤[]y l 1x y l 2x · DX · Y ky l 1y − DY k · X l 2 y l 1y l 2 | · · ·· · · | j DX · Y kxx − DY k · X xx | · · ·m∑l 1 =1· · · |Y ky m · DX − X y m · DY k∣ ∣ ∣ ∣ +y l 1x · ∣∣ ∣ ∣ Yky 1 · DX − X y 1 · DY k | · · ·· · · | j DX · Y kxy l 1 − DY k · X xy l 1 | · · ·m∑l 1 =1 l 2 =1· · · |Yy k · DX − X m y m · DY k∣ ∣ ∣ +m∑y l 1x y l 2x · ∣∣ ∣ Yky · DX − X 1 y 1 · DY k | · · ·· · · | j DX · Y ky l 1y l 2− DY k · X y l 1y l 2 | · · ·· · · |Y ky m · DX − X y m · DY k∣ ∣ ∣ ∣As it is delicate to read, let us say that lines 2, 3 and 4 just express the j-thcolum | j A k | of the determinant |B 1 k| · · · |j A k | · · · |Bm k |, after replacementof A k by its complete expression (5.14).In lines 6, 7, 8; in lines 9, 10, 11; and in lines 12, 13, 14, there arethree families of m×m determinants containing a linear combination (soustraction)having exactly two terms in each column. As in the proof ofLemma 5.10, by multilinarity, we have to develope each such determinant.In principle, for each development, we should get 2 m terms, but since theobtained determinants vanish as soon as the column DY k (modulo a multiplicationby some factor) appears at least twice, it remains only (m+1) nonvanishingdeterminants, those for which the column DY k appears at most.⎥⎦⎤