By Manin Yu.I.

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18. and i s an THEOREM. S3 , t h e n even COROLLARY. S^QN , we have ord[ < Ls > , L]

If i t i s solvable a t a l l , then any solution i s obtained from one solution by addition of any d -constant in 33 . I t i s c l e a r l y solvable if d:33-+33 i s s u r j e c t i v e . This completes the proof. 7. COROLLARY. If the d -constants in 33 coincide with k ( e . g . 6 does not exceed 7W+1 . ( I t w i l l be shown below that in the commutative case i t i s equal to M-\-\, even if d:33-*33 i s not s u r j e c t i v e . ) For matrix rings 33 i t i s possible to obtain an analogous estimate under l e s s r e s t r i c t i v e assumptions which we now axiomatize.

X , i . e . , we We s h a l l f i r s t introduce formal Fourier coefficients oo (•n„|ftgZ) "Vne2Kinx. yvne2lti',x, n U) The Fourier coefficients of the polynomials in u i n f i n i t e l y many v a r i a b l e s B = C[v„] . vn . v±\=mla^otdv±\c±^'0) then obviously . B = © BN i . e . , F(a (y) ) = (2ic/By©n . If BN ord l©^'.. ti°*) = |«i|+• • •+|«*| and i s the space ot polynomials purely of order ( d i r e c t sum). We s e t £? = I I BN • N, The elements of the space OO B are i n f i n i t e s e r i e s 2 fft> /N^BN • Since ord(/£) = o r d / + o r d g for f£BM, g§BN, B N=0 has a natural ring structure: (2/v)(2 ^ ) = 2 A * ' K^/K-MSM ' We call the ring 5((^iJr)) = ( 2 / ^„2xinx fn£B, zc>0, n0>0,v\n\>nQ, ord/„>c|«|} the ring of formal Fourier s e r i e s ; i t has the natural m u l t i p l i c a t i o n \^ifme2,tlm \^hpe2"'px , where hp = y fmgn .