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Any [[ring homomorphism]] <math>\varphi</math> from {{mvar|R}} into a field {{mvar|K}}, defines a ring homomorphism <math>R[\mathbf x] \to K[\mathbf x]</math> (also denoted <math>\varphi</math>), by applying <math>\varphi</math> to the coefficients of the polynomials.
The theorem is: there is an ideal {{mvar|r}} in {{mvar|R}}, uniquely determined by {{mvar|I}}, such that, for every ring homomorphism <math>\varphi</math> from {{mvar|R}} into a field {{mvar|K}}, the homogeneous polynomials <math>\varphi(f_1),\ldots, \varphi(f_k)</math> have a nontrivial common zero (in an algebraic closure of {{mvar|K}}) if and only if <math>\varphi(r)=\{0\}.</math>
Moreover, {{math|1=''r'' = 0}} if {{math|1=''k'' < ''n''}}, and {{mvar|r}} is [[principal ideal|principal]] if {{math|1=''k'' = ''n''}}. In this latter case, a generator of {{mvar|r}} is called the [[Macaulay's resultant|resultant]] of <math>f_1,\ldots, f_k.</math>
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