Let $k$ be an infinite field and $I\subset k [x_1, \ldots ,x_n]$ be a non-zero ideal such that dim $V (I) = q $. Denote by $(f_1; : : : ; s)$ a set of generators of $I$. One can see that in the set $I\cap k [x_{1},...,x_{q+1}]$ there exist non-zero polynomials, depending only on these $q+1$ variables. We aim to bound the minimal degree of the polynomials of this type, and of a Bézout (i.e. membership) relation expressing such a polynomial as a combination of the$f_i$. We also give a relative version of this theorem.