By Manin Yu.I.

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**Extra resources for Algebraic aspects of nonlinear differential equations**

**Example text**

UN-i] + Mv^xUN-NaNv'k+v Below we shall always assume that duN—0). , The operator L i s fixed, while P may vary. 6. THEOREM. If a v , a^_, l i e in the center of 33 and d i s the dimension over k of the space of ^-constants i n 33 , then the dimension of the space of operator P&B\d\, for which ordfP, L\

2) Let a, b£33((£-1)) . We s e t res ( ^ btl') = b_i. Then res(aob — boa)£d93 . I t suffices to verify t h i s for a=Vim, b = u\n . I t f o l l o w s e a s i l y from formula that (1B+B+1) if m+«-f-l>0 cases. and e i t h e r Let us suppose t h a t /ra>0, or m>0, /i<0; [m + n + lj Therefore, rt>0, but mn<0 , and res[a, &]=0 i n t h e remaining t h e second a l t e r n a t i v e i s t r e a t e d a n a l o g o u s l y . Then (m + n + 1)! res[a, b] i s p r o p o r t i o n a l t o valm+n+1)—( l ' \m + n + \}' — l)m*a*1tt'Oi'a+'t+1> and i s a t o t a l derivative by a lemma of Chap.

5 . 10. Proposition. For a l l s>0, S6Q„ and 0 < i < A r — 2 we have <4> ^-M(5-I)=I2(;J(-^-5^^^ Proof. R e l a t i o n s (3) can be c o n s i d e r e d a system of e q u a t i o n s f o r which has t r i a n g u l a r form and can t h e r e f o r e be s o l v e d by i n d u c t i o n . ,v_N+l(s—\), The c l o s e d formulas (4) a r e most e a s i l y o b t a i n e d by w r i t i n g (3) i n o p e r a t o r form tofe&^sQ + doiyv^is-l), where T is the operator for increasing the index by one: T (v^) (5) =v(*l+1 .