Solution to Rubel's question about differentially algebraic dependence on initial values (Q1419648)

The aim of this paper is to answer, in the negative, a question posed by L. A. Rubel, namely that for generic systems of polynomial differential equations, the dependence of the solution on the initial conditions is not differentially algebraic. The term generic has the following meaning: when \(U\in \mathbb{R}^N\) is an open set and \(V\subset U\), then \(V\) is said to be generic on \(U\) if its complement in \(U\) is both of measure \(0\) and of first Baire category. An analytic function \(f(z)\) is called differentially algebraic (DA) if it satisfies some differential equation of the form \[ Q(z, f(z),f'(z),\dots, f^{(n)}(z)= 0, \] for all \(z\) in its domain, where \(Q\) is a nonzero polynomial of \(n+2\) variables. A function of several variables is DA if it is DA in each variables separately. Here, the systems of differential equations \[ y_k'= p_k(y_1,\dots, y_m),\quad k=1,\dots, m,\tag{1} \] with \(p_k\) polynomials are considered, and \(y_k(r_1,\dots, r_m; x)\) denotes the solution of (1) with the initial conditions \[ y_k(0)= r_k,\quad k= 1,\dots, m.\tag{2} \] The question is whether it is true that the dependence of \(y_i\) on \(r_i\) (fixing \(x\) and other initial conditions \(r_k\), \(k\neq j\)) is DA. Let \(S(m,d)\) denote the set of all polynomial systems (1) of size \(m\), where the polynomials \(p_i\) are of degree at most \(d\). For each system \(P\in S(m,d)\) and initial conditions \(r= (r_1,\dots, r_m)\in \mathbb{R}^m\), let \(I(P,r)\) denote the largest interval containing \(0\) for which the solutions of the initial value problem (1), (2), are defined. Let \[ \Lambda(P)= \{(x,r)\mid r\in \mathbb{R}^m,\;x\in I(P,r)\} \] and for each \(1\leq j\leq m\) let \[ \Lambda_j(P)= \{(x, r_1,\dots, r_{j-i}, r_{j+1},\dots, r_m)\mid \exists r;\text{ such that }(x,r_1,\dots, r_m)\in \Lambda(P)\}. \] Assume \(m\geq 2\) and \(d\geq 2\). The main result states that for generic \(P\in S(m, d)\) holds: For any \(1\leq i,\,j\leq m\), and for a generic choice of \((x,r_1,\dots, r_{j-1}, r_{j+1},\dots, r_m)\in \Lambda_j(P)\), the function \(f_{ij}(z)= y_i(r_1,\dots, r_{j-1}, z,r_{j+1},\dots, r_m; x)\) is not DA. Thus, the answer to Rubel's question is very negative.