# Efficient algorithm to prove that a polynomial ideal contains 1

I have the following problem: Suppose to have an ideal $$I\triangleleft k[x_1,...,x_n]$$ defined by generators. There exists an efficient algorithm (perhaps more efficient than calculating the Groebner basis) to determine if $$1\in I$$? More precisely i'm dealing with the following case: $$I=(f_1(x_1,...,x_n),...,f_{n+1}(x_1,...,x_n))$$ so it is in some sense "overdetermined" and is expected to have an empty zero-set. But how can be sure about that without about that without computing Groebner basis which is a very hard problem for high values of $$n$$? Thanks for your time.