586 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



By 5.15, the normal form of FZ is 



(^f4'z) ■ (normal form of i/'fi/'z/''Z). 



Hence the exponent of (p — />„) in tiie /o* diagonal element of the nor- 

 mal form of ^F^zFZ is r — fk . By a similar argument, the exponent 

 of {jp — po) in the A;*^ diagonal element of the normal form of \pzZ is 

 r — £k . Hence, by (e) of 5.1, 



(r - f 'x) + • • • + (r - e[) 



is the exponent of (p — po) in the highest common factor of all ?;-rowed 

 minor determinants of \J/f^zFZ. Similarly 



(r - £1) + • • • + (r - £b) 



is the exponent of (p — po) in the highest common factor of all 6-ro\ved 

 minor determinants of i/'zZ. 



Now yppi^zFZ is a polynomial matrix. A typical 6-rowed minor de- 

 terminant of this matrix is of the form 



^Wz Z ^hN, , (4) 



where the summation is over certain products MbNb of 5-rowed minors 

 Mh of F and 6-rowed minors Kb of Z. For a proof of this, see MacDuffee , 

 Theorem 99.1. The expression (4) is the same as 



E (^pUU){rl^''zNb) (5) 



where the factors (rpz^^b) ai'e now 6-rowed minors of ^zZ. If (p is a factor 

 common to all 6-rowed minors of ^zZ, it certainly is a factor conmion 

 to all expressions (4) or (5). Hence the highest common factor of all 

 6-rowed minor determinants of xpp^pzFZ — i.e., of all expressions (4) or 

 (5), — ^has an exponent for {p — po) no lower than that in the highest 

 common factor of all 6-rowed minor determinants of rpzZ. Hence for 

 any b, 



(r -€[)+■•■ + (r - £6) > (r - fi) + • • . + (r - f,), 



or 



f 1 + • • • + £b > £1 + • • • + f 6 . 



It follows that 



7i + • • • + lb > £i + ■ • • + f 6 . 



This being true for every h, it is certainly true for every h such that all 

 terms on the right are >0 (cf. (2)). This means that for 6 = 1, and for 



