# Algebraic Aspects of Linear Differential and Difference by Hendriks P.A. PDF By Hendriks P.A.

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Extra info for Algebraic Aspects of Linear Differential and Difference Equations

Example text

CCA : 0 0    Ar We denote by (A) the system of di erential equations which corresponds to the matrix A. 1 System (A) is called linear Siegel normal if for any solution f = (f1 ; : : : ; fr )t of (A) , fi = (fi1; : : : ; fini )t and any pi 2 K ni the relation p1 f1 +    + pr fr = 0 implies for each i = 1; : : : ; r that either pi = 0 or fi = 0. 2 System (A) is called homogeneous algebraic Siegel normal if for any N  1 and any solution f = (f1 ; : : : ; fr )t of (A) and any pl ;:::;lr 2 K Nl ;:::;lr X pl ;:::;lr f l ;:::;lr = 0 1 1 l1 ++lr =N 1 1 implies for all r-tuples l1; : : : ; lr with l1 + : : : + lr = N that either pl ;:::;lr = 0 or f l ;:::;lr = 0.

It follows that S 2 Gl(2; k(z)). In particular the Riccati equation has (in nitely many) solutions in k(z). 2 So if k2 = k, then we want to determinepall elds k~  l with [k~ : k]  2. It is obvious that k~ must be of the form k( s), where s is an algebraic integer in l. Recall that l is the splitting eld of the polynomial FH . After a suitable linear substitution z 7! nz , where n 2 Z1 we can assume that FH is monic and that the coecients of FH are algebraic integers. s must be a divisor of the discriminant of this p polynomial.

0 0 ...    Ai;i ;j    0 0 ... 0 ...    An ;n;m  0 B1;1 ... Bi;j ... 9. The 2mn + 1-tuple (K ; ; ; : : : ; Ki;i ;j ; : : : ; Kn;n ;m ; 1)t is a solution of the system (A) consisting of nonzero E -functions. Hence the numbers 1 1 1 K ; ; (1); : : : ; Ki;i ;j (1); : : : ; Kn ;n;m (1); 1 1 1 1 are homogeneous algebraic independent with an e ective measure of homogeneous algebraic independence and so the numbers K ; (1); : : : ; Ki;i (j ); : : : ; Kn;n (m) 1 1 are algebraic independent with an e ective measure of algebraic independence.