Polynomial identities of pre-Lie algebras of vector fields

Polynomial vector fields in $d$ variables form a pre-Lie algebra under directional differentiation. Applying the Amitsur--Levitzki theorem to their Jacobians gives a standard identity of degree $2d+1$. We prove that, over a field of characteristic zero, every identity of degree at most $2d$ follows from the defining pre-Lie identity, and the additional multilinear identities in degree $2d+1$ are spanned by relabellings of the standard identity. The proof uses monomial vector fields to recover coefficients of rooted trees with branching vertices. For $d\geq2$, in degree $2d+2$ we construct identities independent of the consequences of the standard identity, and in dimensions two and three we give complete descriptions using matrix identities and alternating multilinear maps. We also describe how the degree-$(2d+1)$ identities specialize when variables are repeated. A B-series method that has order $p\leq2d+1$ for all autonomous ODEs on $\mathbb R^d$ must satisfy every usual rooted-tree order condition through order~$p$.