Editing Bézout's theorem (section)

==Proofs==
===Using the resultant (plane curves)===
Let {{mvar|P}} and {{mvar|Q}} be two homogeneous polynomials in the indeterminates {{math|''x'', ''y'', ''t''}} of respective degrees {{mvar|p}} and {{mvar|q}}. Their zeros are the [[homogeneous coordinates]] of two [[projective curve]]s. Thus the homogeneous coordinates of their intersection points are the common zeros of {{mvar|P}} and {{mvar|Q}}.

By collecting together the powers of one indeterminate, say {{mvar|y}}, one gets univariate polynomials whose coefficients are homogeneous polynomials in {{mvar|x}} and {{math|t}}.

For technical reasons, one must [[change of coordinates]] in order that the degrees in {{mvar|y}} of {{mvar|P}} and {{mvar|Q}} equal their total degrees ({{mvar|p}} and {{mvar|q}}), and each line passing through two intersection points does not pass through the point {{math|(0, 1, 0)}} (this means that no two point have the same [[Cartesian coordinate system|Cartesian {{mvar|x}}-coordinate]].

The [[resultant]] {{math|''R''(''x'' ,''t'')}} of {{mvar|P}} and {{mvar|Q}} with respect to {{mvar|y}} is a homogeneous polynomial in {{mvar|x}} and {{mvar|t}} that has the following property:  <math>R(\alpha,\tau)=0</math> with <math>(\alpha, \tau)\ne (0,0)</math> if and only if it exist <math>\beta</math> such that <math>(\alpha, \beta, \tau)</math> is a common zero of {{mvar|P}} and {{mvar|Q}} (see {{slink|Resultant|Zeros}}). The above technical condition ensures that <math>\beta</math> is unique. The first above technical condition  means that the degrees used in the definition of the resultant are {{mvar|p}} and {{mvar|q}}; this implies that the degree of {{mvar|R}} is {{mvar|pq}} (see {{slink|Resultant|Homogeneity}}).

As {{mvar|R}} is a homogeneous polynomial in two indeterminates, the [[fundamental theorem of algebra]] implies that {{mvar|R}} is a product of {{mvar|pq}} linear polynomials. If one defines the multiplicity of a common zero of {{mvar|P}} and {{mvar|Q}} as the number of occurrences of the corresponding factor in the product, Bézout's theorem is thus proved.

For proving that the intersection multiplicity that has just been defined equals the definition in terms of a deformation, it suffices to remark that the resultant and thus its linear factors are [[continuous function]]s of the coefficients of {{mvar|P}} and {{mvar|Q}}.

Proving the equality with other definitions of intersection multiplicities relies on the technicalities of these definitions and is therefore outside the scope of this article.

===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.

===Using the degree of an ideal===
Bézout's theorem can be proved by recurrence on the number of polynomials 
by using the following theorem.

''Let {{mvar|V}} be a [[projective algebraic set]] of [[dimension of an algebraic variety|dimension]] <math>\delta</math> and [[degree of an algebraic variety|degree]] <math>d_1</math>, and {{mvar|H}} be a hypersurface (defined by a single polynomial) of degree <math>d_2</math>, that does not contain any [[irreducible component]] of {{mvar|V}}; under these hypotheses, the intersection of {{mvar|V}} and {{mvar|H}} has dimension <math>\delta-1</math> and degree <math>d_1d_2.</math>''

For a (sketched) proof using [[Hilbert series]], see {{slink|Hilbert series and Hilbert polynomial|Degree of a projective variety and Bézout's theorem}}.

Beside allowing a conceptually simple proof of Bézout's theorem, this theorem  is fundamental for [[intersection theory]], since this theory is essentially devoted to the study of intersection multiplicities when the hypotheses of the above theorem do not apply.