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ふぁい f such that f(X) = 0 holds in R[X] iff φふぁい f holds in R; φふぁい f is of course the formula saying that all the coefficients of f(X) vanish. Moreover, φふぁい 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ふぁい(a) in , such that φふぁい(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ふぁい(y) in this sense exist and can be taken as infinite disjunctions of positive existential formulas.