Resultants and the Hermite Normal Form

1 Introduction 1
1.1 Polynom ials .......................................................................................... 1
1.2 R esultants ............................................................................................. 3
1.3 Multivariate polynomials ................................................................. 6
1.4 Symmetric fu n ction s ........................................................................ 7
1.5 The Resultant in terms of the ro o ts ............................................ 9
1.6 Two resultant resu lts ........................................................................ 11
1.6.1 The discrim inant ................................................................ 11
1.6.2 Cyclotomic polynom ials .................................................. 13
1.7 Subresultants ...................................................................................... 17
2 Equivalence of m atrices 23
2.1 Definitions ............................................................................................. 23
2.2 One-sided equivalence ..................................................................... 25
2.3 Two-sided eq u ivalen ce ....................................................................... 28
3 The Integral polynom ial range 32
3.1 Introduction .......................................................................................... 32
3.2 Preliminaries ...................................................................................... 33
3.3 The Bigradient, Gp ............................................................................ 34
3.4 The Infinite Bigradient, G0 0 .......................................................... 39
3.5 Examples and corollaries ................................................................. 46
3.6 Monic polynomials and the resultant m a trix ............................ 51
3.7 A bound for the stability of G ^ ................................................... 59
4 R elated Topics 61
4.1 The Rational Polynomial R ange ................................................... 61
4.2 Matrix rings and multivariate polynom ials .............................. 63
Bibliography 66