Algebraic aspects of nonlinear differential equations by Manin Yu.I.

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By Manin Yu.I.

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S>0, ord [ < I* > , k . Suppose t h a t Then for any — 2, o r d P < A f UN, »JV-I l i e i n t h e c e n t e r of 38, t h a t M>0 w is invertible, a l l w , t h a t t h e s e t of t h e space of o p e r a t o r s P£33[d] , s ttN d - c o n s t a n t s i n 33 s Indeed, s i n c e t h e o r d e r of t h e o p e r a t o r sN If a l l i e s i n t h e L] i s e x a c t l y sN, and i t s l e a d i n g g e n e r a t e a space of dimension coefficient M-\-l .

Nonzero elements sites Let cfck S3Q be commutative, on t h e d i a g o n a l . 8 a ) , b ) , and c) a r e s a t i s f i e d . semisimple m a t r i x of Mt(k) ct = Cj. It is This i m p l i e s t h a t any i s a l s o semisimple i n our s e n s e of t h e word. As i s known, t h e converse i s also t r u e . 8 we d e n o t e by d t h e dimension of KetdnSB* over k . 10. THEOREM. If UN i s i n v e r t i b l e , c o n s t a n t , and semisimple and dimension of t h e space of o p e r a t o r s i s p r e c i s e l y equal to d(M+\) Proof.

R6$[d] be a finite family of operators. We call it independent if the sum of 33 -modules 2 33LPl... if/ is direct. We write LP=LPt... L"/ and |/>|=2 pj . 3. LEMMA. ,S=$T=S) the module. ,Lr , and the rank of a submodule does not exceed the rank of is an independent family of operators, then r < « . ,Lr, Proof. There exists a constant /, depending on the degrees of the operators such that for all m > 0 there is the following imbedding of free ^-modules: £i 3hlPa \p]

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