A derivation on a commutative ring R is a function ∂:R → R satisfying universally

• ∂(x + y) = ∂(x) + ∂(y)
• ∂(x y) = x ∂(y) + y ∂ (x)

A differential field with n commuting derivations is a field K given together with n derivations ∂₁, …, ∂_n for which

For example, one might take for K the field of meromorphic functions on Cⁿ and ∂_i = ∂/∂ z_i.

The theory DF_0,n of differential fields of characteristic zero with n commuting derivations has a model completion, the theory DCF_0,n of differentially closed fields of characteristic zero with n commuting derivations.

In a differential field (K, ∂₁, …, ∂_n), the constant field is the field C := { x ∈ K | (&forall i) ∂_i(x) = 0 }.

A differential polynomial in l variables over K is a polynomial (in the sense of universal algebra) in l variables over K considered in the language L(+, ×, ∂₁, …, ∂_n).