This paper is devoted to study the Lipschitzian/Holderian type global error bound for systems of many finitely convex polynomial inequalities over a polyhedral constraint. Firstly, for systems of this type, we show that under a suitable asymtotic qualification condition, the Lipschitzian type global error bound property is equivalent to the Abadie qualification condition, in particular, the Lipschitzian type global error bound is satisfied under the Slater condition. Secondly, without regularity conditions, the H$\ddot{\mbox{o}}$lderian global error bound with an implicit exponent is invertigated.