FORMAL REALlZAHlLirV THEORY II 



08 1 



r = sup {—m, 0) 



is called the order of (he polo of R{p) at p^ , even if r = 0. 



5.1 Let Z{p) be au n X n matrix wiiose elements Zrs(p) are rational 

 functions of the complex variable />. We write 



Nrsip) 



Zrsip) = 



DrAp)' 



wiiere .Vrs and Dr.. are n^latixcly piime polynomials. Let ""^/.(p) b(> the 

 hvist common multiple of all Drs{p), (1 < '", s < n), so normalized that 

 tiie coefficient of the highest power of p appearing in ^z(p), (the leading 

 coefficient) is unity. Then ^z(p) is uniquely determined by Z{p). 



The matrix ^z(p)Z(p) has polynomial elements. Its Smith normal 

 form is a diagonal matrix E(p), 



E{p) = 



'Ey{p) 



Eiiv) 



0" 



Eniv) 



•0 



= A{p)-^,{p)Z{p)B{p), (1) 



with the following properties: 



(a) R is the rank of ^z(p)Z(p). 



(b) Each Ei(p), I < i < R, is a polynomial with unit leading coef- 

 ficient. 



(c) Each Ei{p) is a factor of Ei+i(p), 1 < i < R — I. 



(d) A(p) and B(p) are polynomial matrices, each with a constant 

 non- vanishing determinant. 



(e) Ei(p)E2{p) • • • Ek(p) is the normalized (and therefore unique) 

 highest common factor of all /v-rowed minor determinants of 

 ^z{p)Z(p). 



These properties of E{p) are developed for example, in Bocher^^, 

 Theorems 2 and 3 of paragraph 91 and Theorem 1 of paragraph 94. A 

 simple variation of this last cited theorem will also pro\'e the following 

 uniqueness lemma. 



.■3.11 Lemma: If some E (p) satisfies (1) and (a), (b), (c) and (d) above, 

 all written with superscripts on each E, and on A and B, then E^{p) = 

 E{p). 



Proof: E (p) is equivalent to E(p) in the sense of paragraph 94 of 



