Etude algebrique et algorithmique des singularites des by Hubert E.

By Hubert E.

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7 Proposition: Let R be a di erential ring. Whenever R contains a eld isomorphic to Q , the radical of a di erential ideal of R is a di erential ideal, and thus a radical di erential ideal. 2 . For an element a of R and an integer we show by induction that for any r, 0  r  , 1, Proof: Consider any derivation 2 N qr =  , 1 : : :  , rp ,r,1  This is true for r = 0 since q0 = p ,1 p = p Assume qr 2 p for 0  r , 1. Then qr = p2r+1 2 p 2p :  , 1 : : :  , r , 1p ,r,2 p2r+1 +2r + 1  , 1 : : :  , rp ,r,1 p2r 2p 2 p and therefore q qr = qr+1 +2r +1qr 2p 2 p ; which drives us to the desired conclusion.

T. every other element of A. Such a set is nite and triangular : no pair of di erential polynomials in A have the same leader. t. t. any element of A. Let A be an auto-reduced set. t. 9 . t. A satisfying 1. When A is an auto-reduced set of FfY g, we will note mathbfhA the product of all initials and separants of the di erential polynomial in A. Characteristic sets A ranking on FfY g induces a pre-order on the set of all auto-reduced subsets of FfY g. Let A = a1 ; : : : ; ar and B = b1 ; : : : ; bs be two auto-reduced subsets.

40 d . 1 y~1 x et y~2 x, pour c = 0 et a; b = 0; 0;  13 ; 13 ;  31 ; , 241  Exemple : Soit l'
equation di erentielle 6 y y00 , 5 y023 + 729 y4 = 0: dont l'unique solution singuli ere est y0x = 0. La solution g
en
erale peut ^etre donn
ee par 3 y2x = x , a + b x , a2 o u a et b sont les constantes arbitraires. y0x = 0 est donc une enveloppe des courbes de la solution g
en
erale et le contact est d'ordre 3. Nous repr
esentons quelques unes de ces solutions. 1. 02 a = ,1; 0; 1, b = 1 2 42 d .