Differential Algebra Preliminaries

We assume that everything, which is undefined, is as in [Ritt (1950] and in [Kolchin (1973)]. Let $\delta$ be a derivation operator and let R be a differential ring, i.e. a commutative ring with unit and a derivation $\delta$ acting on it, such that the field of rational numbers ${\bf Q}$ is contained in R.

A differential field F is a field, which is also a differential ring. Let C_R={$a \in R$: $\delta(a)=0$} be the ring of constants of R and let ${\bf N}_{0}=\{0, 1, \ldots, n, \ldots\}$.

REMARK 1   ${\bf Q} \subseteq C_{R}$ because ${\bf Q} \subseteq R$.

EXAMPLE 1   R= $C^{\infty}({\bf R})$, R= ${\bf Q}(t)$, R= ${\bf R}(t)$ and R=field of the meromorphic functions on a domain of ${\bf C}$ with the usual derivation $\delta$= $\frac{\partial}{\partial t}$ are differential rings. R= $C^{m}({\bf R})$ for all $m\in {\bf N}_{0}$ is not a differential ring.

If $a \in {\bf N}_{0}$, then order of $\delta^{a}$ is ord( $\delta^{a}$)=a.

DEFINITION 1   S= $R\{ y_{1}, \ldots, y_{m} \}$=R[ $\delta^{n}y_{i}$: $i=1, \ldots, m$ and $n \in {\bf N}_{0}$] is the differential ring of the differential polynomials in the differential indeterminates $y_{1}, \ldots, y_{m}$ with coefficients in the differential ring R.

If $f \in S$, then order of f is ord(f)=max{ $n \in {\bf N}_{0}$: f contains a power product in $y_{1},\ldots,y_{m},\delta y_{1},\ldots,\delta y_{m},\ldots, \delta^{n}y_{1},\ldots,\delta^{n}y_{m}$ with nonzero coefficient }.

DEFINITION 2   An ideal I of S is a differential ideal iff $\delta^{n}(s) \in I$ for all $n \in {\bf N}_{0}$ and all $s \in I$. If $A \subseteq S$, then [A]=( $\delta^{n}(s)$: $s \in A$, $n \in {\bf N}_{0}$ ) denotes the smallest differential ideal containing A.

Let R be a differential ring of functions in the independent variable t, let $\delta$= $\frac{\partial}{\partial t}$ and let y=y(t) be a differential indeterminate. y⁽ⁿ⁾= $\delta^{n}y$ is the derivative of order n of ywith respect to t for all positive integers n, while y⁽⁰⁾= $\delta^{0}y$=y.

So an ordinary algebraic differential equation in m dependent variables $y_{1}, \ldots, y_{m}$ with coefficients in R is nothing else than the equality $\{ f=0 \}$ for some differential polynomial f in S.