Basic global relative invariants for nonlinear differential equations
著者
書誌事項
Basic global relative invariants for nonlinear differential equations
（Memoirs of the American Mathematical Society, no. 888）
American Mathematical Society, 2007
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注記
"November 2007, volume 190, number 888 (first of three numbers)."
Includes bibliographical references (p.357-358) and index
内容説明・目次
内容説明
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.
目次
Part 1. Foundations for a General Theory: Introduction The coefficients $c_{i,j}^{*}(z)$ of (1.3) The coefficients $c_{i,j}^{**}(\zeta)$ of (1.5) Isolated results needed for completeness Composite transformations and reductions Related Laguerre-Forsyth canonical forms Part 2. The Basic Relative Invariants for $Q {m} = 0$ when $m\geq 2$: Formulas that involve $L_{i,j}(z)$ Basic semi-invariants of the first kind for $m \geq 2$ Formulas that involve $V_{i,j}(z)$ Basic semi-invariants of the second kind for $m \geq 2$ The existence of basic relative invariants The uniqueness of basic relative invariants Real-valued functions of a real variable Part 3. Supplementary Results: Relative invariants via basic ones for $m \geq 2$ Results about $Q {m}$ as a quadratic form Machine computations The simplest of the Fano-type problems for (1.1) Paul Appell's condition of solvability for $Q {m} = 0$ Appell's condition for $Q {2} = 0$ and related topics Rational semi-invariants and relative invariants Part 4. Generalization for $H_{m, n} = 0$: Introduction to the equations $H_{m, n} = 0$ Basic relative invariants for $H_{1,n} = 0$ when $n \geq 2$ Laguerre-Forsyth forms for $H_{m, n} = 0$ when $m \geq 2$ Formulas for basic relative invariants when $m \geq 2$ Extensions of Chapter 7 to $H_{m,n} = 0$, when $m \geq 2$ Extensions of Chapter 9 to $H_{m,n} = 0$, when $m \geq 2$ Basic relative invariants for $H_{m, n} = 0$ when $m \geq2$ Additional Classes of Equations: The class of equations specified by $y""(z)$$y'(z)$ Formulations of greater generality Invariants for simple equations unlike (29.1) Bibliography Index.
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