Electronic ISBN:  9781470404949 
Product Code:  MEMO/190/888.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 190; 2007; 365 ppMSC: Primary 34;
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 123year 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 \(n\)thdegree 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

Chapters

Part 1. Foundations for a general theory

1. Introduction

2. The coefficients $c^*_{i,j}(z)$ of (1.3)

3. The coefficients $c^{**}_{i,j}(\zeta )$ of (1.5)

4. Isolated results needed for completeness

5. Composite transformations and reductions

6. Related LaguerreForsyth canonical forms

Part 2. The basic relative invariants for $Q_m=0$ when $m\ge 2$

7. Formulas that involve $L_{i,j}(z)$

8. Basic semiinvariants of the first kind for $m \geq 2$

9. Formulas that involve $V_{i,j}(z)$

10. Basic semiinvariants of the second kind for $m \geq 2$

11. The existence of basic relative invariants

12. The uniqueness of basic relative invariants

13. Realvalued functions of a real variable

Part 3. Supplementary results

14. Relative invariants via basic ones for $m \geq 2$

15. Results about $Q_m$ as a quadratic form

16. Machine computations

17. The simplest of the Fanotype problems for (1.1)

18. Paul Appell’s condition of solvability for $Q_m = 0$

19. Appell’s condition for $Q_2 = 0$ and related topics

20. Rational semiinvariants and relative invariants

Part 4. Generalizations for $H_{m,n}=0$

21. Introduction to the equations $H_{m,n} = 0$

22. Basic relative invariants for $H_{1,n} = 0$ when $n \geq 2$

23. LaguerreForsyth forms for $H_{m,n} = 0$ when $m \geq 2$

24. Formulas for basic relative invariants when $m \geq 2$

25. Extensions of Chapter 7 to $H_{m,n} = 0$, when $m \geq 2$

26. Extensions of Chapter 9 to $H_{m,n} = 0$, when $m \geq 2$

27. Basic relative invariants for $H_{m,n} = 0$ when $m \geq 2$

Part 5. Additional classes of equations

28. The class of equations specified by $y”(z) y’(z)$

29. Formulations of greater generality

30. Invariants for simple equations unlike (29.1)


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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 123year 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 \(n\)thdegree 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.

Chapters

Part 1. Foundations for a general theory

1. Introduction

2. The coefficients $c^*_{i,j}(z)$ of (1.3)

3. The coefficients $c^{**}_{i,j}(\zeta )$ of (1.5)

4. Isolated results needed for completeness

5. Composite transformations and reductions

6. Related LaguerreForsyth canonical forms

Part 2. The basic relative invariants for $Q_m=0$ when $m\ge 2$

7. Formulas that involve $L_{i,j}(z)$

8. Basic semiinvariants of the first kind for $m \geq 2$

9. Formulas that involve $V_{i,j}(z)$

10. Basic semiinvariants of the second kind for $m \geq 2$

11. The existence of basic relative invariants

12. The uniqueness of basic relative invariants

13. Realvalued functions of a real variable

Part 3. Supplementary results

14. Relative invariants via basic ones for $m \geq 2$

15. Results about $Q_m$ as a quadratic form

16. Machine computations

17. The simplest of the Fanotype problems for (1.1)

18. Paul Appell’s condition of solvability for $Q_m = 0$

19. Appell’s condition for $Q_2 = 0$ and related topics

20. Rational semiinvariants and relative invariants

Part 4. Generalizations for $H_{m,n}=0$

21. Introduction to the equations $H_{m,n} = 0$

22. Basic relative invariants for $H_{1,n} = 0$ when $n \geq 2$

23. LaguerreForsyth forms for $H_{m,n} = 0$ when $m \geq 2$

24. Formulas for basic relative invariants when $m \geq 2$

25. Extensions of Chapter 7 to $H_{m,n} = 0$, when $m \geq 2$

26. Extensions of Chapter 9 to $H_{m,n} = 0$, when $m \geq 2$

27. Basic relative invariants for $H_{m,n} = 0$ when $m \geq 2$

Part 5. Additional classes of equations

28. The class of equations specified by $y”(z) y’(z)$

29. Formulations of greater generality

30. Invariants for simple equations unlike (29.1)