Contents

Chapter 1. Introduction 1

1.1. General summary 1

1.2. The ring 7£m of polynomials to be employed throughout 2

1.3. Terminology used throughout (except as modified in Chapter 12) 3

1.4. Principal results 5

1.5. Symbolism for polynomials in 7£m 7

1.6. Miscellaneous observations 8

1.7. Order of presentation 9

Chapter 2. Some Problems of Historical Importance 12

2.1. The older semi-invariants 12

2.2. A challenging problem posed by J. Liouville in 1839 13

2.3. The first construction of a decisive set for any m 3 16

2.4. Decisive sets of semi-invariants 18

2.5. Awkward formulations involving the older semi-invariants 19

Chapter 3. Illustrations for Some Results in Chapters 1 and 2 21

3.1. {G2, ..., Gm} and {G3, ..., G

m

} are not decisive sets 21

3.2. A simple check on the consistency of (1.20)-(1.27) 22

3.3. A simple check on the consistency of (1.12)—(1.15) 23

3.4. Two types of symbolic sums and their evaluation 24

3.5. Computations when m is a symbol for an integer 3 27

3.6. A comprehensive check on the consistency of (1.8)-(1.27) 28

Chapter 4. Ln and In^ as Semi-Invariants of the First Kind 31

Chapter 5. Vn and Jn^ as Semi-Invariants of the Second Kind 34

Chapter 6. The Coefficients of Transformed Equations 39

6.1. Alternative formulas for c**(C) in (1.5) 39

6.2. The coefficients of a composite transformation 40

6.3. Several examples 44

6.4. Proof of an old observation 45

6.5. Conditions for transformed equations 46

6.6. Formulas for later reference 49

Chapter 7. Formulas That Involve Ln(z) or In,n(z) 50

7.1. The coefficients of (6.8) when di{() = d2(() = 0 50

7.2. Derivatives for the coefficients of (6.8) when d\(Q = ^(C) = 0 53

7.3. Identities for the coefficients of (6.8) when di(C) = Gfe(C) = 0 55