Let f : C n ! C be any polynomial function. By using global polar methods, we i n troduce models for the bers of f and we study the monodromy at atypical values of f , including the value in nity. W e construct a geometric monodromy w i t h c o n trolled behavior and de ne global relative monodromy with respect to a general linear form. We prove localization results for the relative monodromy and derive a zeta-function formula for the monodromy around an atypical value. We compute the relative zeta function in several cases and emphasize the di erences to the \classical" local situation.

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