# Basic Global Relative Invariants for Nonlinear Differential Equations

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*Roger Chalkley*

The problem of deducing the basic relative
invariants possessed by monic homogeneous linear differential
equations of order \(m\) was initiated in 1879 with Edmund
Laguerre's success for the special case \(m = 3\). It was
solved in number 744 of the Memoirs of the AMS (March 2002), by a
procedure that explicitly constructs, for any \(m \geq3\), each
of the \(m - 2\) basic relative invariants. During that
123-year time span, only a few results were published about the basic
relative invariants for other classes of ordinary differential
equations.

With respect to any fixed integer \(\,m \geq 1\), the author begins
by explicitly specifying the basic relative invariants for the class
\(\,\mathcal{C}_{m,2}\) that contains equations like \(Q_{m}
= 0\) in which \(Q_{m}\) is a quadratic form in \(y(z),
\, \dots, \, y^{(m)}(z)\) having meromorphic coefficients written
symmetrically and the coefficient of \(\bigl( y^{(m)}(z)
\bigr)^{2}\) is \(1\). Then, in terms of any fixed
positive integers \(m\) and \(n\), the author explicitly specifies
the basic relative invariants for the class
\(\,\mathcal{C}_{m,n}\) that contains equations like
\(H_{m,n} = 0\) in which \(H_{m,n}\) is an
\(n\)th-degree form in \(y(z), \, \dots, \, y^{(m)}(z)\)
having meromorphic coefficients written symmetrically and the
coefficient of \(\bigl( y^{(m)}(z) \bigr)^{n}\) is
\(1\). These results enable the author to obtain the basic relative
invariants for additional classes of ordinary differential
equations.

#### Table of Contents

# Table of Contents

## Basic Global Relative Invariants for Nonlinear Differential Equations

- Contents iii4 free
- Preface ix10 free
- Part 1. Foundations for a General Theory 114 free
- Chapter 1. Introduction 316
- 1.1. Historical motivation 316
- 1.2. Context and definitions used throughout Chapters 1–20 417
- 1.3. Main Theorem 821
- 1.4. Notational abbreviations employed in Chapters 7–10 1023
- 1.5. Illustrations for the use of formulas (1.18) (1.28) when ra ≥ 2 1023
- 1.6. Completion of Paul Appell's research in [7] about Q[sub(2)] = 0 1326
- 1.7. Inclusion of homogeneous linear differential equations 1730
- 1.8. Order of presentation 2033

- Chapter 2. The Coefficients c*[sub(i,j)](z) of (1.3) 2336
- Chapter 3. The Coefficients c**[sub(i,j)](ζ) of (1.5) 2942
- Chapter 4. Isolated Results Needed for Completeness 3548
- 4.1. Nonsolutions of nontrivial equations 3548
- 4.2. Semi-invariants of the second kind are isobaric 3750
- 4.3. Substitutions in regard to the derivation ' for S[sub(m)] 3851
- 4.4. All of the relative invariants for (1.1) when m = 1 3952
- 4.5. Isobaric semi-invariants of weight 2 when m ≥ 2 4356
- 4.6. Further semi-invariants of the second kind when m ≥ 2 4558

- Chapter 5. Composite Transformations and Reductions 4760
- 5.1. The composite of substitutions (1.2) and (1.4) 4760
- 5.2. The condition d[sub(0,1)](ζ) ≡ d[sub(0,2)](ζ) ≡ 0 for (5.1) when m ≥ 2 4962
- 5.3. Laguerre-Forsyth canonical forms for linear equations 5063
- 5.4. A Laguerre-Forsyth canonical form for (1.1) when m ≥ 2 5164
- 5.5. There are no relative invariants in Q{w[sub(0,1)],w[sub(0,2)]} 5265

- Chapter 6. Related Laguerre-Forsyth Canonical Forms 5366

- Part 2. The Basic Relative Invariants for Q[sub(m)] = 0 when ra ≥)] 2 6780
- Chapter 7. Formulas That Involve L[sub(i,j)](z) 6982
- Chapter 8. Basic Semi-Invariants of the First Kind for m ≥2 87100
- Chapter 9. Formulas That Involve V[sub(i,j)](z) 93106
- Chapter 10. Basic Semi- Invariants of the Second Kind for m ≥ 2 111124
- Chapter 11. The Existence of Basic Relative Invariants 119132
- Chapter 12. The Uniqueness of Basic Relative Invariants 121134
- Chapter 13. Real-Valued Functions of a Real Variable 135148

- Part 3. Supplementary Results 141154
- Chapter 14. Relative Invariants via Basic Ones for m ≥ 2 143156
- Chapter 15. Results about Qm as a Quadratic Form 157170
- Chapter 16. Machine Computations 167180
- 16.1. Expansion of D[sub(2)] in terms of a[sub(m,2)], I[sub(m,1,1)], I[sub(m,1,2)], and I[sub(m,2,2)] 167180
- 16.2. The expansions for E[sub(6)] and E[sub(7)] in (1.81) and (1.82) 168181
- 16.3. A comprehensive check on the consistency of (1.14)–(1.38) 169182
- 16.4. The relative invariants of weight ≤ 9 for the equations Q[sub(2)] = 0 172185
- 16.5. Entry of keyboard instructions 177190

- Chapter 17. The Simplest of the Fano-Type Problems for (1.1) 179192
- Chapter 18. Paul Appell's Condition of Solvability for Q[sub(m)] = 0 185198
- Chapter 19. Appell's Condition for Q[sub(2)] = 0 and Related Topics 193206
- 19.1. Consequences of Chapter 18 for the equations Q[sub(2)] = 0 193206
- 19.2. An improvement for Proposition 19.6 197210
- 19.3. An example to illustrate Theorem 19.7 198211
- 19.4. Other forms for the nonsingular solutions in Theorem 19.7 199212
- 19.5. Conditions of the type (u[sub(1)](z))[sup(2)] – 4 u[sub(0)](z) u[sub(2)](z) [omitted] 0 201214
- 19.6. Equations constructed to have given nonsingular solutions 203216
- 19.7. Absence of movable branch points 206219
- 19.8. Two results for third- order linear equations 208221
- 19.9. Extensions to linear equations of higher order 210223
- 19.10. Linear substitutions in binary forms 213226
- 19.11. Properties of the polynomial T[sub(n)] in (19.120) 216229

- Chapter 20. Rational Semi-Invariants and Relative Invariants 219232
- 20.1. Terminology for an extended context 219232
- 20.2. The integer s in Definition 20.2 220233
- 20.3. A context for the remainder of this chapter 223236
- 20.4. A technical construction needed for Section 20.5 225238
- 20.5. Rational semi-invariants of the first kind 230243
- 20.6. A technical construction needed for Section 20.7 234247
- 20.7. Rational semi-invariants of the second kind 240253
- 20.8. The structure of rational relative invariants 243256
- 20.9. Substitutions into rational functions of Q[sub(m)] 244257

- Part 4. Generalizations for H[sub(m,n)] = 0 247260
- Chapter 21. Introduction to the Equations H[sub(m,n)] = 0 249262
- 21.1. Transformations produced by changing the variables in H[sub(m,n)] = 0 249262
- 21.2. Context and definitions 251264
- 21.3. A summary of results and a derivation 'for S[sub(m,n)] 253266
- 21.4. Inclusion of relative invariants for H[sub(m,q)] = 0 when 1 ≤ q ≤ n 253266
- 21.5. Nonsolutions of nontrivial equations 255268

- Chapter 22. Basic Relative Invariants for H[sub(1,n)] = 0 when n ≥ 2 257270
- Chapter 23. Laguerre- Forsyth Forms for H[sub(m,n)] = 0 when m ≥ 2 265278
- 23.1. The composite of substitutions (21.2) and (21.4) 265278
- 23.2. Laguerre-Forsyth reductions when m ≥ 2 267280
- 23.3. Related Laguerre-Forsyth canonical forms 268281
- 23.4. Identities for the coefficients of related canonical forms 271284
- 23.5. Use of computer algebra to check Theorem 23.8 272285
- 23.6. Properties of a[sub(m,2)] in (23.14) and 6[sub(m,2)] in (23.16) 273286

- Chapter 24. Formulas for Basic Relative Invariants when m ≥ 2 275288
- Chapter 25. Extensions of Chapter 7 to H[sub(m,n)] = 0, when m ≥ 2 279292
- Chapter 26. Extensions of Chapter 9 to H[sub(m,n)] = 0, when m ≥ 2 293306
- Chapter 27. Basic Relative Invariants for H[sub(m,n)] = 0 when m ≥ 2 307320
- 27.1. Preliminary results 307320
- 27.2. Principal results 314327
- 27.3. Some polynomials that are not relative invariants 316329
- 27.4. Uniqueness of basic relative invariants 316329
- 27.5. The basic relative invariant of index (l[sub(1)],...,l[sub(n)] when l[sub(n)] = 1 317330
- 27.6. The number of basic relative invariants 318331
- 27.7. Relative invariants via basic ones for m ≥ 2 320333
- 27.8. Rational semi-invariants for various classes of equations 321334

- Part 5. Additional Classes of Equations 323336
- Chapter 28. The Class of Equations Specified by y''(z) y'(z) 325338
- 28.1. Notation and terminology 325338
- 28.2. Principal formulas 326339
- 28.3. The relative invariants of weight ≤ 9 for the equations (28.1) 328341
- 28.4. Computational procedure for Section 28.3 330343
- 28.5. Laguerre-Forsyth reductions for the equations (28.1) 331344
- 28.6. All of the relative invariants for the equations (28.1) 332345

- Chapter 29. Formulations of Greater Generality 335348
- Chapter 30. Invariants for Simple Equations unlike (29.1) 347360

- Bibliography 357370
- Index 359372 free