Given a polynomial ODE in $n$-dimensions of maximal degree $d$ $$ \dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n $$ we define an algebraic solution as an any curve $\gamma: [t_0,t_1] \rightarrow \mathbb{R}^n$ solving the ODE and fulfilling $$ G(\gamma(t))=\sum_{i_{1},\dots,i_{n}=1}^{d'}b_{i_{1},\dots,i_{n}}{\gamma_1(t)}^{i_{1}}\dots {\gamma_n(t)}^{i_{n}}=0 \quad \forall t \in [t_0,t_1] $$ for some real coefficients $b_{i_1,...,i_n}$.
I am interested in the following set of questions as well as hints to the literature (or partial results in lower dimensions):
- Is there an algorithm to generally determine whether a polynomial DS has an algebraic solution?
- Is there a bound on how many algebraic solutions a polynomial ODE can have depending on $n,d$? (I know that some polynomial ODEs like $\dot{x}=-y, \dot{y}=x$ have infinitely many algebraic solutions, but if there are only finite ones, can we bound them.)
- Can we write those polynomial ODEs in a general form? (I know for $n=2$, two algebraic curves $f,g$ and an antisymmetric matrix $S$, $F=S \nabla (gf)$ has $f,g$ as invariant algebraic curve. But I am not sure if this is the general form.)
- How many polynomial ODEs generally allow for algebraic solutions? (E.g. identifying the polynomial vector fields with their coefficients which form the set $\mathbb{R}^{n \cdot \binom{n+d}{n}}$. Are the coefficients which allow an algebraic solution dense in $\mathbb{R}^{n \cdot \binom{n+d}{n}}$ or some open subset.)
