Editing Bézout's theorem (section)

===Using {{mvar|U}}-resultant===
{{main|Resultant#Uresultant|l1={{mvar|U}}-Resultant}}
In the early 20th century, [[Francis Sowerby Macaulay]] introduced the [[multivariate resultant]] (also known as ''Macaulay's resultant'') of {{mvar|n}} [[homogeneous polynomial]]s in {{mvar|n}} indeterminates, which is  generalization of the usual [[resultant]] of two polynomials. Macaulay's resultant is a polynomial function of the coefficients of {{mvar|n}} homogeneous polynomials that is zero if and only the polynomials have a nontrivial (that is some component is nonzero) common zero in an [[algebraically closed field]] containing the coefficients.

The {{mvar|U}}-resultant is a particular instance of Macaulay's resultant, introduced also by Macaulay. Given {{mvar|n}} homogeneous polynomials <math>f_1,\ldots,f_n</math> in {{math|''n'' + 1}} indeterminates <math>x_0, \ldots, x_n,</math> the {{mvar|U}}-resultant is the resultant of <math>f_1,\ldots,f_n,</math> and <math>U_0x_0+\cdots +U_nx_n,</math> where the coefficients <math>U_0, \ldots, U_n</math> are auxiliary indeterminates. The {{mvar|U}}-resultant is a homogeneous polynomial in <math>U_0, \ldots, U_n,</math> whose degree is the product of the degrees of the <math>f_i.</math>

Although a multivariate polynomial is generally [[irreducible polynomial|irreducible]], the {{mvar|U}}-resultant can be factorized into linear (in the <math>U_i</math>) polynomials over an [[algebraically closed field]] containing the coefficients of the <math>f_i.</math> These linear factors correspond to the common zeros of the <math>f_i</math> in the following way: to each common zero <math>(\alpha_0, \ldots, \alpha_n)</math> corresponds a linear factor <math>(\alpha_0 U_0 + \cdots + \alpha_n U_n),</math> and conversely.

This proves Bézout's theorem, if the multiplicity of a common zero is defined as the multiplicity of the corresponding linear factor of the {{mvar|U}}-resultant. As for the preceding proof, the equality of this multiplicity with the definition by deformation results from the continuity of the {{mvar|U}}-resultant as a function of the coefficients of the <math>f_i.</math>

This proof of Bézout's theorem seems the oldest proof that satisfies the modern criteria of rigor.