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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.

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