# Basic Global Relative Invariants for Homogeneous Linear Differential Equations

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*Roger Chalkley*

Given any fixed integer \(m \ge 3\), we present simple formulas for \(m - 2\) algebraically independent polynomials over \(\mathbb{Q}\) having the remarkable property, with respect to transformations of homogeneous linear differential equations of order \(m\), that each polynomial is both a semi-invariant of the first kind (with respect to changes of the dependent variable) and a semi-invariant of the second kind (with respect to changes of the independent variable). These relative invariants are suitable for global studies in several different contexts and do not require Laguerre-Forsyth reductions for their evaluation. In contrast, all of the general formulas for basic relative invariants that have been proposed by other researchers during the last 113 years are merely local ones that are either much too complicated or require a Laguerre-Forsyth reduction for each evaluation. Unlike numerous studies of relative invariants from 1888 onward, our global approach completely avoids infinitesimal transformations and the compromised rigor associated with them. This memoir has been made completely self-contained in that the proofs for all of its main results are independent of earlier papers on relative invariants. In particular, rigorous proofs are included for several basic assertions from the 1880's that have previously been based on incomplete arguments.

#### Table of Contents

# Table of Contents

## Basic Global Relative Invariants for Homogeneous Linear Differential Equations

- Contents vii8 free
- Chapter 1. Introduction 114 free
- 1.1. General summary 114
- 1.2. The ring R[sub(m)] of polynomials to be employed throughout 215
- 1.3. Terminology used throughout (except as modified in Chapter 12) 316
- 1.4. Principal results 518
- 1.5. Symbolism for polynomials in R[sub(m)] 720
- 1.6. Miscellaneous observations 821
- 1.7. Order of presentation 922

- Chapter 2. Some Problems of Historical Importance 1225
- Chapter 3. Illustrations for Some Results in Chapters 1 and 2 2134
- 3.1. {G[sub(2)], …, G[sub(m)]} and {G[sub(3)], …, G[sub(m)]} are not decisive sets 2134
- 3.2. A simple check on the consistency of (1.20)–(1.27) 2235
- 3.3. A simple check on the consistency of (1.12)–(1.15) 2336
- 3.4. Two types of symbolic sums and their evaluation 2437
- 3.5. Computations when m is a symbol for an integer ≥ 3 2740
- 3.6. A comprehensive check on the consistency of (1.8)–(1.27) 2841

- Chapter 4. L[sub(n)] and I[sub(n,i)] as Semi-Invariants of the First Kind 3144
- Chapter 5. V[sub(n)] and J[sub(n,i)] as Semi-Invariants of the Second Kind 3447
- Chapter 6. The Coefficients of Transformed Equations 3952
- Chapter 7. Formulas That Involve L[sub(n)](z) or I[sub(n,n)](z) 5063
- Chapter 8. Formulas That Involve V[sub(n)](z) or J[sub(n,n)](z) 6174
- Chapter 9. Verification of I[sub(n,n)] ≡ J[sub(n,n)]and Various Observations 7184
- Chapter 10. The Local Constructions of Earlier Research 7790
- Chapter 11. Relations for G[sub(i)], H[sub(i)], and L[sub(i)] That Yield Equivalent Formulas for Basic Relative Invariants 8194
- 11.1. The identity H[sub(i)] ≡ G[sub(i)] D[sub(i,m i)] 8194
- 11.2. Formulas for L[sub(0)], …, L[sub(m)] in terms of G[sub(0)], …, G[sub(m)] 8396
- 11.3. Formulas for L[sub(3)], …, L[sub(m)] in terms of H[sub(3)], …, H[sub(m)] 87100
- 11.4. Formulas for G[sub(0)], …, G[sub(m)] in terms of L[sub(0)], …, L[sub(m)] 90103
- 11.5. Formulas for H[sub(3)], …, H[sub(m)] in terms of L[sub(3)], …, L[sub(m)] 93106

- Chapter 12. Real-Valued Functions of a Real Variable 94107
- Chapter 13. A Constructive Method for Imposing Conditions on Laguerre-Forsyth Canonical Forms 104117
- Chapter 14. Additional Formulas for K[sub(i,j)], U[sub(i,j)], A[sub(i,j)], D[sub(i,j)], … 108121
- 14.1. Alternative formulas for K[sub(i,j)] in (1.12)–(1.14) 108121
- 14.2. Alternative formulas for U[sub(i,j)] in (1.21)–(1.23) 109122
- 14.3. Alternative formulas for A[sub(i,j)] in (2.19)–(2.21) 110123
- 14.4. Alternative formulas for D[sub(i,j)] in (2.42)–(2.44) 110123
- 14.5. Alternative formulas for β[sub(i,j)](z) in (6.2) (6.3) 111124
- 14.6. Alternative formulas for φ[sub(i,j)](z) in (6.10) (6.11) 112125
- 14.7. Polynomials Q[sub(i,j)] and R[sub(i)] for use in Section 15.2 112125

- Chapter 15. Three Canonical Forms Are Now Available 114127
- 15.1. Reduction of (1.1) to a Halphen canonical form 114127
- 15.2. A neglected canonical form for any (1.1) having m ≥ 2 116129
- 15.3. Semi-invariants R[sub(2)], …, R[sub(m)] analogous to G[sub(2)], …, G[sub(m)] 119132
- 15.4. {R[sub(2)], …, R[sub(m)]} and {R[sub(3)], …, R[sub(m)]} are not decisive sets 121134
- 15.5. Semi-invariants S[sub(3)], …, S[sub(m)] analogous to H[sub(3)], …, H[sub(m)] 121134
- 15.6. The identity S[sub(i)] ≡ R[sub(i)] T[sub(i,m i)] 122135
- 15.7. Solving (14.28) for ω[sub(i)] in terms of R[sub(0)], , R[sub(m)] 125138

- Chapter 16. Interesting Problems that Require Further Study 130143
- Appendix A. Results Needed for Self- Containment 133146
- A.1. The coefficients c*[sub(i)](z) for (1.3) 134147
- A.2. The coefficients c**[sub(i)](ζ) for (1.5) 135148
- A.3. Semi-invariants of the second kind given by S[sup((1))] + kb[sub(1)]S 138151
- A.4. Non-solutions for non-zero equations 139152
- A.5. Semi-invariants of the second kind are isobaric 140153
- A.6. Supplementary observations about invariants 142155
- A.7. Identities relating the coefficients of (1.3) or (1.5) to (1.1) 146159
- A.8. Pencil-and-paper computation for I[sub(3,3)] 148161
- A.9. Machine computations for the coefficients C[sub(i)](z) of (15.32) 149162

- Appendix B. Related Studies for a Class of Nonlinear Equations 151164
- Appendix C. Polynomials That Are Linear in a Key Variable 156169
- Appendix D. Rational Semi-Invariants and Relative Invariants 165178
- D.1. Introduction 165178
- D.2. Definitions of rational semi-invariants and relative invariants 166179
- D.3. The integer s in Definition D.2 166179
- D.4. A context for the remainder of this appendix 169182
- D.5. A technical construction needed for Section D.6 172185
- D.6. Rational semi-invariants of the first kind 176189
- D.7. A technical construction needed for Section D.8 180193
- D.8. Rational semi-invariants of the second kind 185198
- D.9. The structure of rational relative invariants 188201
- D.10. The structure of absolute invariants 188201
- D.11. Substitutions into fractions of Q[sub(m)] 189202

- Appendix E. Generating Additional Relative Invariants 192205
- Bibliography 197210
- Index 200213 free