The purpose of this note is to prove the following theorem: Solutions common to two distinct components' of the manifold of a difference polynomial annul the separants of the polynomial. We begin by considering a field I, not necessarily a difference field, and a set of polynomials F,, F2,, * * *, Fp in K[ul, * , u.; xl, * *I* xp], the ui and xj being indeterminates, where for each j,lj=1, * ,p-1, Fj is free of the xk, k >j. We shall show that any zero of F,, * * , Fp which annuls no formal partial derivative OFj1/Ox belongs to just one component of { F,, , Fp }o.2 Furthermore, this component is of dimension q. PROOF. Let ui='yi, i=1, ... , q; xj=acj, j=1, * , p, be a zero of F,, Fp which annuls no 9Fj/Oxj. If yl . * ,yqa'; aI, a is a zero of F,, , Fp which specializes to yi, I, yq; ail, * , apt then this zero too annuls no OFi/Oxi. It follows from this that a' is algebraic over K (-y', , 'y), and that for each k, 1