Introduction

Let t be an independent variable and let y=y(t) be a dependent variable. Let y⁽ⁿ⁾ be the derivative of order n of y with respect to t. Let

$$P=P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k})$$

be a polynomial in the variables $y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots,v_{k}$ with coefficients in a differential field F of functions, as example $F={\bf R}(t)$ field of rational functions with coefficients in ${\bf R}$.

Let z_i=exp(Q_i) and let v_j=log(R_j), where $Q_{i}=Q_{i}(y,y^{(1)},\ldots,y^{(q_{i})})$ and $R_{j}=R_{j}(y,y^{(1)},\ldots,y^{(r_{j})})$ are polynomial functions with coefficients in the same differential field F for all $i=1,\ldots,h$ and all $j=1,\ldots,k$.
The ordinary 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 in general nonlinear and nonalgebraic.

The equation $\{P=0\}$ is equivalent to the system

{ $P=P(y,y^{(1)},\ldots,y^{(n)},exp(Q_{1}),\ldots,exp(Q_{h}),log(R_{1}), \ldots,log(R_{k}))=0$,
P⁽¹⁾=0, ..., P^(m)=0, z_i=exp(Q_i), v_j=log(R_j), z_i⁽¹⁾=Q_i⁽¹⁾exp(Q_i)=Q_i⁽¹⁾z_i, R_jv_j⁽¹⁾=R_j⁽¹⁾, $i=1,\ldots,h$, $j=1,\ldots,k$, $m\in {\bf N}_{0}$},

where P^(m) is the derivative of order m of the function P=P(t) with respect to t for all $m \geq 1$.

Let $P^{(1)}=P(y,y^{(1)},\ldots,y^{(n)},z_{1},\ldots,z_{h},v_{1},\ldots, v_{k})^{(1)... ...1}^{(1)},\ldots,z_{h}^{(1)},v_{1},\ldots,v_{k},v_{1}^{(1)},\ldots, v_{k}^{(1)})$.

By algebraically eliminating 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

$$P_{1}'=P_{1}'(y,y^{(1)},\ldots,y^{(N+1)},z_{1},\ldots, z_{h},v_{1},\ldots,v_{k},v_{1}^{(1)},\ldots,v_{k-1}^{(1)})$$

is obtained, where N=max{n,q₁,...,q_h, r₁,...,r_k}.

By algebraically eliminating v_k between P and P₁' a polynomial $P_{1}=P_{1}(y,y^{(1)},\ldots,y^{(N+1)}, z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-1})$ is obtained.

By using this procedure we can eliminate $z_{1},\ldots,z_{h},v_{1},\ldots,v_{k-1}$ and we obtain a polynomial P_h+k only in y and its derivatives, such that every solution of $\{P=0\}$ is a solution of $\{P_{h+k}=0\}$.

The same procedure can be applied to dynamical systems.

There are many differential elimination procedures. Some of them are based on the characteristic set theory of Wu-Ritt [Ritt (1950,Kolchin (1973)]
[Wu (1987),Wu (1989),Wang (1993)]; other procedures use in addition the Gröbner bases theory for polynomial ideals [Boulier (1994),Boulier (1996)] [Boulier and Lazard and Ollivier and Petitot (1995),Diop (1991)]
[Diop (1991),Diop (1991),Diop (1992),Diop and Fliess (1991)]
[Fliess (1989),Fliess (1990)], some others are based on the differential resultants theory [Carra' Ferro (1997),Carra' Ferro (1997)] and the differential Gröbner bases theory for differential polynomial ideals [Carra' Ferro (1987),Ollivier (1990),Weispfenning (1993)].