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The quantization of non-Abelian gauge theories is known to be plagued byGribov copies. Typical examples are the copies related to zero modes of theFaddeev-Popov operator, which give rise to singularities in the ghostpropagator. In this work we present an exact and compact expression for theghost propagator as a function of external gauge fields, in SU(N) Yang-Millstheory in the Landau gauge. It is shown, to all orders, that the condition forthe ghost propagator not to have a pole, the so-called Gribov's no-polecondition, can be implemented by demanding a nonvanishing expectation value fora functional of the gauge fields that turns out to be Zwanziger's horizonfunction. The action allowing to implement this condition is theGribov-Zwanziger action. This establishes in a precise way the equivalencebetween Gribov's no-pole condition and Zwanziger's horizon condition.

An all-order proof of the equivalence between Gribov's no-pole and Zwanziger's horizon conditions