Electronic ISBN:  9781470403379 
Product Code:  MEMO/156/744.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 156; 2002; 204 ppMSC: Primary 34;
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 semiinvariant of the first kind (with respect to changes of the dependent variable) and a semiinvariant 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 LaguerreForsyth 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 LaguerreForsyth 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 selfcontained 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.
ReadershipGraduate students and research mathematicians interested in ordinary differential equations.

Table of Contents

Chapters

1. Introduction

2. Some problems of historical importance

3. Illustrations for some results in Chapters 1 and 2

4. $L_n$ and $I_{n,i}$ as semiinvariants of the first kind

5. $V_n$ and $J_{n,i}$ as semiinvariants of the second kind

6. The coefficients of transformed equations

7. Formulas that involve $L_n(z)$ or $I_{n,n}(z)$

8. Formulas that involve $V_n(z)$ or $J_{n,n}(z)$

9. Verification of $I_{n,n} \equiv J_{n,n}$ and various observations

10. The local constructions of earlier research

11. Relations for $G_i$, $H_i$, and $L_i$ that yield equivalent formulas for basic relative invariants

12. Realvalued functions of a real variable

13. A constructive method for imposing conditions on LaguerreForsyth canonical forms

14. Additional formulas for $K_{i,j}$, $U_{i,j}$, $A_{i,j}$, $D_{i,j}$, …

15. Three canonical forms are now available

16. Interesting problems that require further study


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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 semiinvariant of the first kind (with respect to changes of the dependent variable) and a semiinvariant 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 LaguerreForsyth 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 LaguerreForsyth 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 selfcontained 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.
Graduate students and research mathematicians interested in ordinary differential equations.

Chapters

1. Introduction

2. Some problems of historical importance

3. Illustrations for some results in Chapters 1 and 2

4. $L_n$ and $I_{n,i}$ as semiinvariants of the first kind

5. $V_n$ and $J_{n,i}$ as semiinvariants of the second kind

6. The coefficients of transformed equations

7. Formulas that involve $L_n(z)$ or $I_{n,n}(z)$

8. Formulas that involve $V_n(z)$ or $J_{n,n}(z)$

9. Verification of $I_{n,n} \equiv J_{n,n}$ and various observations

10. The local constructions of earlier research

11. Relations for $G_i$, $H_i$, and $L_i$ that yield equivalent formulas for basic relative invariants

12. Realvalued functions of a real variable

13. A constructive method for imposing conditions on LaguerreForsyth canonical forms

14. Additional formulas for $K_{i,j}$, $U_{i,j}$, $A_{i,j}$, $D_{i,j}$, …

15. Three canonical forms are now available

16. Interesting problems that require further study