An Elimination Procedure of One Differential Variable

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

Let $P^{(1)}=P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots, v_{k})^{(1)... ...1}^{(1)},\ldots,z_{h}^{(1)},v_{1},\ldots,v_{k},v_{1}^{(1)},\ldots, v_{k}^{(1)})$ and let J₁ be the ideal (P, P⁽¹⁾, z_i⁽¹⁾-Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾-R_j⁽¹⁾, $i=1,\ldots,h$, $j=1,\ldots,k$) in the polynomial ring $R[y,\ldots,y^{(N)}, y^{(N+1)},z_{1},\ldots,z_{h},z_{1}^{(1)},\ldots,z_{h}^{(1)},v_{1},\ldots,v_{k}, v_{1}^{(1)},\\ \ldots,v_{k}^{(1)}]$.

Let $\sigma_{1}$ be the lexicographic term ordering on the set of power products in $\{y,\ldots,y^{(N+1)},z_{1},\ldots,z_{h},z_{1}^{(1)}, \ldots,z_{h}^{(1)},v_{1},\ldots,v_{k},v_{1}^{(1)},\ldots,v_{k}^{(1)}\}$, with $y<\ldots<y^{(N)}<y^{(N+1)}<z_{1}<z_{1}^{(1)}<\ldots<z_{h}< z_{h}^{(1)}<v_{1}<v_{1}^{(1)}<\ldots<v_{k}<v_{k}^{(1)}$.

Let G₁ be the reduced Gröbner basis of the ideal J₁ with respect to $\sigma_{1}$
[Buchberger (1976),Buchberger (1976)]. Since P⁽¹⁾= $\frac{\partial}{\partial v_{k}^{(1)}}(P^{(1)})v_{k}^{(1)}+T_{1}$ and R_kv_k⁽¹⁾-R_k⁽¹⁾ have degree one in v_k⁽¹⁾, then P and S₁= $R_{k}T_{1}+\frac{\partial}{\partial v_{k}^{(1)}}(P^{(1)}) R_{k}^{(1)}$ are in $J_{1} \cap R[y,\ldots,y^{(N+1)},z_{1},\ldots, z_{h},z_{1}^{(1)},\ldots,z_{h}^{(1)},v_{1},\ldots,v_{k},v_{1}^{(1)}, \ldots,v_{k-1}^{(1)}]$. So it is possible to eliminate v_k⁽¹⁾between P⁽¹⁾ and R_kv_k⁽¹⁾-R_k⁽¹⁾ and by using equalities z_i⁽¹⁾=Q_i⁽¹⁾z_i a polynomial P₁'= $P_{1}'(y,y^{(1)},\ldots,y^{(N+1)},z_{1},\ldots, z_{h},v_{1},\ldots,v_{k},\\ v_{1}^{(1)},\ldots,v_{k-1}^{(1)})$ is obtained with $P_{1}' \neq P$.

If v_k does not appear in P₁', then P₁' is in $J_{1} \cap R[y,\ldots,y^{(N+1)},z_{1},\ldots,z_{h},\\ v_{1},\ldots,v_{k-1},v_{1}^{(1)},\ldots,v_{k-1}^{(1)}]$. If v_k appears in P₁', then $1 \leq deg_{v_{k}}P_{1}'\leq deg_{v_{k}}P$ and the resultant of P and P₁' with respect to the variable v_k is in $J_{1} \cap F[y,\ldots,y^{(N+1)},z_{1},\ldots,z_{h},z_{1}^{(1)}, \ldots,z_{h}^{(1)},v_{1},\ldots,v_{k-1},v_{1}^{(1)},\ldots,v_{k-1}^{(1)}]$. So there exists at least one differential polynomial P₁ in $G_{1} \cap F[y,\ldots,y^{(N+1)},z_{1},\ldots,\\ z_{h},z_{1}^{(1)}, \ldots,z_{h}^{(1)},v_{1},\ldots,v_{k-1},v_{1}^{(1)},\ldots,v_{k-1}^{(1)}]$by definition of Gröbner basis
with respect to a lexicographic term ordering.

Let G₁₁= $G_{1} \cap R[y,\ldots,y^{(N+1)},z_{1},\ldots,z_{h},z_{1}^{(1)}, \ldots,z_{h}^{(1)},v_{1},\ldots,v_{k-1},v_{1}^{(1)},\\ \ldots,v_{k-1}^{(1)}]$. Since G₁₁ is the reduced Gröbner basis of J₁₁= $J_{1} \cap R[y,\ldots,y^{(N+1)},\\ z_{1},\ldots,z_{h},z_{1}^{(1)}, \ldots,z_{h}^{(1)},v_{1},\ldots,v_{k-1},v_{1}^{(1)},\ldots,v_{k-1}^{(1)}]$with respect to $\sigma_{1}$ and
$z_{i}^{(1)}-Q_{i}^{(1)}z_{i}, R_{j}v_{j}^{(1)}-R_{j}^{(1)} \in J_{11}$ for all $i=1,\ldots,h$, $j=1,\ldots,k-1$, then $P_{1} \in G_{11}$ and P₁= $P_{1}(y,\ldots, y^{(N+1)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-1}, v_{1}^{(1)},\ldots,v_{k-1}^{(1)})$. If deg_vk-1⁽¹⁾P₁=d, then the pseudoremainder P₁^* of P₁ with respect to R_k-1v_k-1⁽¹⁾-R_k-1⁽¹⁾ as polynomials in v_k-1⁽¹⁾ is in J₁₁ and v_k-1⁽¹⁾ does not appear in P₁^*. So we can assume that P₁= $P_{1}(y,\ldots,y^{(N+1)},z_{1},\ldots,z_{h},v_{1}, \ldots,v_{k-1},v_{1}^{(1)},\ldots,v_{k-2}^{(1)})$.

REMARK 2   The elimination of the differential variable v_k in the system $(\beta)$can be also obtained in the following ways.

(i) By using the Ritt process of differential reduction for the extended characteristic set, it is sufficient to find the extended characteristic set of the differential polynomials Pand R_kv_k⁽¹⁾-R_k⁽¹⁾. It riquires the pseudo division of P⁽¹⁾ by R_kv_k⁽¹⁾-R_k⁽¹⁾ as polynomials in $R[y,\ldots,y^{(N+1)},z_{1},\ldots,z_{h},z_{1}^{(1)}, \ldots,\\ z_{h}^{(1)},v_{1},\ldots,v_{k},v_{1}^{(1)},\ldots,v_{k-1}^{(1)}] [v_{k}^{(1)}]$ and the extended characteristic set of the partial remainder of the above pseudodivision R₁ and P [Ritt (1950].

(ii) By using the differential resultant theory, e.g. by finding the differential resultant of the differential polynomials P and R_kv_k⁽¹⁾-R_k⁽¹⁾, as differential polynomials in $R\{ y,z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-1}\}\{v_{k}\}$. By differential resultant theory such differential resultant is equal to the resultant of the polynomials P, P⁽¹⁾ and R_kv_k⁽¹⁾-R_k⁽¹⁾ in the polynomial ring $R[y,\ldots,y^{(N+1)},z_{1},\ldots,z_{h},z_{1}^{(1)},\\ \ldots,z_{h}^{(1)},v_{1},\ldots,v_{k-1},v_{1}^{(1)},\ldots,v_{k-1}^{(1)}] [v_{k},v_{k}^{(1)}]$ [Carra' Ferro (1997)].
In all such elimination procedures of one differential variable the differential polynomials P, P⁽¹⁾ and R_kv_k⁽¹⁾-R_k⁽¹⁾ are involved.

(iii) By using the differential Gröbner basis theory with respect to an elimination differential term ordering [Carra' Ferro (1987),Ollivier (1990)]
[Weispfenning (1993)].

REMARK 3   The same elimination procedures hold if the variable z_h has to be eliminated in the system $(\beta)$. It is sufficient to work with the differential polynomials P, P⁽¹⁾ and z_h⁽¹⁾-Q_h⁽¹⁾z_h.