Let R be a commutative ring with 1 and R[X 1, …, Xn ] the polynomial ring in n variables over R. Then for any relation f(X) = 0 in R[X] there exists a conjunction of equations φ f such that f(X) = 0 holds in R[X] iff φ f holds in R; φ f is of course the formula saying that all the coefficients of f(X) vanish. Moreover, φ f is independent of R and formed uniformly for all polynomials f up to a given formal degree. In this paper we investigate first order theories T for which a similar phenomenon holds. More precisely, we let TAH be the universal Horn part of a theory T and look at free extensions of models of T in the class of models of TAH . We ask whether an atomic relation t 1(X, a) = t 2(X, a) or R(t 1(X, a), …, tn (X, a)) in can be equivalently expressed by a finite or infinitary formula φ(a) in , such that φ(y) depends only on t i{X, y) and not on or a 1, …, am ∈ A.

We will show that for a wide class of theories T “defining formulas” φ(y) in this sense exist and can be taken as infinite disjunctions of positive existential formulas.