Let f1,. . . , fn be a collection of polynomials in d complex variables z1,. . . ,zd. Suppose that it is possible to find another collection of polynomials g1,. . . , gn such that
f1(z)g1(z) + f2(z)g2(z) + . . . + fn(z)gn(z) = 1
for every complex d-tuple z = (z1,. . . , zd). Then it follows immediately that no such d-tuple can be a root of every single ft, since otherwise the left-hand side would equal 0. Remarkably, the converse also holds: that is, if there is no d-tuple for which the polynomials fi all vanish simultaneously, then it is possible to find polynomials gi such that the above identity holds. This result is known as the weak Nullstellensatz.
A short (but clever) argument can be used to deduce Hilbert’s Nullstellensatz from the weak Nullstellensatz. This again is a statement where a condition that is obviously necessary turns out to be sufficient. Suppose that h is another polynomial in d complex variables, that r is a positive integer, and that the polynomial hr can be written in the form f1g1 + f2g2 + . . . + fngn for some collection of polynomials g1,. . . , gn. It follows immediately that h(z) = 0 whenever fi(Z) = 0 for every i. Hilbert’s Nullstellensatz states that if h(z) = 0 whenever fi(z) = 0 for every i, then there must be some positive integer r and some collection of polynomials g1,. . . , gn such that hr = f1g1 + f2g2 + . . . + fngn.
Hilbert’s Nullstellensatz is discussed further in ALGEBRAIC GEOMETRY [IV.4 §§5, 12].