FORMAL KKALIZ AlilLirV I'llKoKY 11 585 



5.25 Dcfniilion: roiisidci- fwo nintri('(>s Z(/)) mid Zi(p), with 



.S{Z, /h) = [71 , 7.. , ■ • • , 7„|, 



N(Z, , •/>„) = [71 , 72 , • • • , tVI- 

 V^'v say 



S(Z, po) > SiZ, , p,) (2) 



if and only if 



Ti + 72 + • • • + 7a- > 7i + 72 + • ■ • + 7I 

 for ('\-(My /,■ = 1,2, • • • ,11. We say 



S{Z, p,) = S(Z, , /;o) (3) 



if 



7/c = Ih 



for A- = 1, 2, • • • , /). It is easy to see that (3) is equi\^alent to the simul- 

 taneous validity of (2) and the reverse inequality. 



5.2G Theorem: Let pn ^ oo be a pole of Z(p). Let F(p) be a rational 

 n X // matrix which is finite at /Ai • Tlien 



SiZ, po) > SiFZ, po). 



In paiiiculai', if F(p) is also non-singular at po , then 



S{Z, Po) = S(FZ, Po). 



Proof: Let \Pf(p) and rpzip) be the least common denominators of 

 the elements of F{p) and Z(p), respectively. Then the exponent of 

 {p — Po) in ypz(p) is r, while in 4^f(p) it is zero by 5.23. 



Let —Ck be the exponent of (p — po) in the k^^ diagonal element of 

 the normal form of Z, and — f^ the similar (juantity for FZ. Then 



ii > f2 > • ■ ■ > f „ , 



£l > £2 > • ■ • > t'n , 



by (c') of 5.13. Let 



7/,- = sup (fk , 0), 



yi = sup {i-'k , 0), 

 Then 7^ > Ck- , 7t > ^^ , and 



S{Z, Po) = [71 , 72 , • • ■ , 7.1, 



S(FZ,po) -\y[,y'., ■■■ ,y'A. 



