An Elimination Procedure of Finitely Many Differential Variables

Let $(\eta)$ be the system of ordinary algebraic differential equations:

$(\eta)$={ P= $P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k})$=0, z_i⁽¹⁾-Q_i⁽¹⁾z_i=0,
R_jv_j⁽¹⁾-R_j⁽¹⁾=0, $i=1,\ldots,h$, $j=1,\ldots,k$ }.

with $P \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k} \}$, $Q_{i}, R_{j} \in F\{ y \}$, ord(Q_i)=q_i and
ord(R_j)=r_j for all i and j.

Let N=max{n,q_i,r_j: $i=1,\ldots,h$, $j=1,\ldots,k$}.

By using an elimination procedure of the variable v_k the system $(\eta)$ becomes equivalent to the system

$(\eta_{1})$={ P=0, P⁽¹⁾=0, P₁=0, z_i⁽¹⁾-Q_i⁽¹⁾z_i=0,
R_jv_j⁽¹⁾-R_j⁽¹⁾=0, $i=1,\ldots,h$, $j=1,\ldots,k$ }.

with $P \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k} \}$, $P_{1} \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-1} \}$, Q_iand R_j in $F\{ y \}$, ord(P₁)=N+1, ord(Q_i)=q_i and ord(R_j)=r_j for all i and j.

By using an elimination procedure of the variable v_k-1 the system $(\eta_{1})$becomes equivalent to the system

$(\eta_{2})$={ P=0, P⁽¹⁾=0, P₁=0, P₁⁽¹⁾=0, P₂=0, z_i⁽¹⁾-Q_i⁽¹⁾z_i=0,
R_jv_j⁽¹⁾-R_j⁽¹⁾=0, $i=1,\ldots,h$, $j=1,\ldots,k$ }.

with $P \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k} \}$, $P_{1} \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-1} \}$,
$P_{2} \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-2} \}$, $Q_{i}, R_{j} \in F\{ y \}$, ord(P₁)=N+1, ord(P₂)=
N+2, ord(Q_i)=q_i and ord(R_j)=r_j for all i and j.

Finally by using recursively an elimination procedure of the variables $v_{k-2},\ldots,v_{1},z_{h},\ldots,z_{h-i}$ the system $(\eta)$ is equivalent to the system

$(\eta_{k+i})$={ P=0, P⁽¹⁾=0, P₁=0, P₁⁽¹⁾=0, ..., P_k=0, P_k⁽¹⁾=0,..., P_k+i-1=0, P_k+i-1⁽¹⁾=0, P_k+i=0, z_i⁽¹⁾-Q_i⁽¹⁾z_i=0, R_jv_j⁽¹⁾-R_j⁽¹⁾=0, $i=1,\ldots,h$, $j=1,\ldots,k$ }.

with $P \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k} \}$, $P_{1} \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-1} \}$,...,
$P_{k} \in F\{ y,z_{1},\ldots,z_{h} \}$,..., $P_{k+i-1} \in F\{ y,z_{1},\ldots,z_{h-i} \}$, $P_{k+i} \in F\{ y,z_{1},\ldots,\\ z_{h-i-1} \}$, $Q_{i}, R_{j} \in F\{ y \}$, ord(P₁)=N+1,..., ord(P_k)=N+k,...,
ord(P_k+i-1)=N+k+i-1, ord(P_k+i)=N+k+i, ord(Q_i)=q_i and ord(R_j)=r_j for all i and j.

If i=h, then the system $(\eta)$ is equivalent to the system

$(\eta_{k+h})$={ P=0, P⁽¹⁾=0, P₁=0, P₁⁽¹⁾=0, ..., P_k=0, P_k⁽¹⁾=0,..., P_k+h-1=0, P_k+h-1⁽¹⁾=0, P_k+h=0, z_i⁽¹⁾-Q_i⁽¹⁾z_i=0, R_jv_j⁽¹⁾-R_j⁽¹⁾=0, $i=1,\ldots,h$, $j=1,\ldots,k$ }.

with $P \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k} \}$, $P_{1} \in F\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-1} \}$,...,
$P_{k} \in F\{ y,z_{1},\ldots,z_{h} \}$,..., $P_{k+h-1} \in F\{ y,z_{1} \}$, $P_{k+h} \in F\{ y \}$, $Q_{i}, R_{j} \in F\{ y \}$, ord(P₁)=N+1,..., ord(P_k)=N+k, ..., ord(P_k+h-1)=N+k+h-1,
ord(P_k+h)=N+k+h, ord(Q_i)=q_i and ord(R_j)=r_j for all i and j.

Now the differential ideal I=[P, z_i⁽¹⁾-Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾-R_j⁽¹⁾, $i=1,\ldots,h$, $j=1,\ldots,k$]= [P, P₁=0,..., P_k,...,P_k+h-1, P_k+h=0, z_i⁽¹⁾-Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾-R_j⁽¹⁾, $i=1,\ldots,h$, $j=1,\ldots,k$] so the systems $(\beta)$ and $(\eta)$ are equivalent.

Let $(\gamma')$ be the following system

$(\gamma')$={ P=0, P⁽¹⁾=0, P₁=0, P₁⁽¹⁾=0, ..., P_k=0, P_k⁽¹⁾=0,...,
P_k+h-1=0, P_k+h-1⁽¹⁾=0, P_k+h=0, z_i= exp(Q_i), v_j= log(R_j),
z_i⁽¹⁾-Q_i⁽¹⁾z_i=0, R_jv_j⁽¹⁾-R_j⁽¹⁾=0, $i=1,\ldots,h$, $j=1,\ldots,k$ }.

The system $(\gamma')$ is equivalent to the system $(\alpha)$.

Since each elimination of one variable among z_i and v_j riquires one more derivative of P by 3.1.1., then $P_{1},\ldots,P_{h+k} \in I$ and the system $(\gamma')$ is equivalent to the following system $(\gamma)$:

$(\gamma)$={ P=0, P⁽¹⁾=0,..., P^(h+k)=0, z_i= exp(Q_i), v_j= log(R_j),
z_i⁽¹⁾-Q_i⁽¹⁾z_i=0, R_jv_j⁽¹⁾-R_j⁽¹⁾=0, $i=1,\ldots,h$, $j=1,\ldots,k$ }.

Finally the differential equation { $P=P(y,y^{(1)},\ldots,y^{(n)},exp(Q_{1}),\ldots,exp(Q_{h}),log(R_{1}), \ldots,log(R_{k}))=0$ } is equivalent to the system

$(\alpha')$={ P= $P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k})$=0, P_h+k=
$P_{h+k}(y,y^{(1)},\ldots,y^{(N+h+k)})$= 0, z_i= exp(Q_i), v_j= log(R_j),
$i=1,\ldots,h$, $j=1,\ldots,k$, },

REMARK 4   The elimination of the differential variables $z_{1},\ldots,z_{h}, v_{1},\ldots,\\ v_{k}$ in the system $(\beta)$ can be also obtained in the following ways.

(i) By using the Ritt's theory of characteristic sets it is sufficient to find the unique differential polynomial $P_{h+k} \in R\{y\}$ in the extended characteristic set of the set of differential polynomials $\{$ P, z_i⁽¹⁾-Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾-R_j⁽¹⁾: $l=1,\ldots,h+k$, $i=1,\ldots,h$, $j=1,\ldots,k$$\}$ with respect to a ranking O of the differential variables, such that $y<_{O}<z_{1}<_{O}\ldots<_{O}z_{h}<_{O}v_{1}<_{O}\ldots<_{O}v_{k}$.

(ii) By using the Gröbner bases techniques, it is sufficient to find the Gröbner basis G of the ideal J_h+k=( $P,P^{(l)}, z_{i}^{(1)}-Q_{i}^{(1)}z_{i},(z_{i}^{(1)}-Q_{i}^{(1)}z_{i})^{(l)},\\ R_{j}v_{j}^{(1)}-R_{j}^{(1)},(R_{j}v_{j}^{(1)}-R_{j}^{(1)})^{(l)}$: $l=1,\ldots,h+k$, $i=1,\ldots,h$, $j=1,\ldots,k$) in $R[y,\ldots,y^{(N+h+k)},z_{1},\ldots,z_{1}^{(h+k)},\ldots,z_{h},\ldots, z_{h}^{(h+k)},v_{1},\ldots,v_{1}^{(h+k)},\ldots,v_{k},\ldots,\\ v_{k}^{(h+k)}]$with respect to the lexicographic term ordering $\sigma$ with $y<_{\sigma}\ldots<_{\sigma}y^{(N+h+k)}<_{\sigma}z_{1}<_{\sigma}\ldots <_{\sigma... ...1}^{(h+k} <_{\sigma}\ldots<_{\sigma}v_{k}<_{\sigma}\ldots<_{\sigma}v_{k}^{(h+k}$. The riquired differential polynomial P_h+k is in $G \cap R[y,\ldots,y^{(N+h+k)}]$.

(iii) By using the differential resultant theory, e.g. by finding the differential resultant of the h+k+1 differential polynomials P, $z_{1}^{(1)}-Q_{1}^{(1)}z_{1},\ldots,z_{h}^{(1)}-Q_{h}^{(1)}z_{h}$, $R_{1}v_{1}^{(1)}-R_{1}^{(1)},\ldots,R_{k}v_{k}^{(1)}-R_{k}^{(1)}$, as differential polynomials in $R\{y\}\{z_{1},\\ \ldots,z_{h},v_{1},\ldots, v_{k-1},v_{k}\}$.
Since ord(P)=n, ord(Q_i)=q_i and ord(R_j)=r_j for all i and j, let N^*= $n+ \sum_{i=1,\ldots,h}max(1,q_{i})+\sum_{j=1,\ldots,k}max(1,r_{j})$. By differential resultant theory such differential resultant is equal to the resultant of the polynomials P,...,
P^(N*-ord(P)), $z_{i}^{(1)}-Q_{i}^{(1)}z_{i},\ldots, (z_{i}^{(1)}-Q_{i}^{(1)}z_{i})^{(N^{*}-max... ...)}-R_{j}^{(1)},\ldots,\\ (R_{j}v_{j}^{(1)}-R_{j}^{(1)})^{(N^{*}-max(1,r_{j}))}$, $i=1,\ldots,h$, $j=1,\ldots,k$) in the polynomial ring R[ $y,\ldots,y^{(N^{*})},z_{i},\ldots, z_{i}^{(N^{*}-max(1,q_{i}))},v_{j},\ldots,v_{j}^{(N^{*}-max(1,r_{j}))}]$: $i=1,\ldots,h$, $j=1,\ldots,k$].
Since each elimination process of the above differential variables riquires at most h+k derivatives of P, z_i⁽¹⁾-Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾-R_j⁽¹⁾, then this elimination process holds also when $P \in C^{r+h+k}({\bf R)}$ for some $r \in {\bf N}_{0}$.