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Bézout's theorem

Bézout's theorem is a central result in stating that two curves of degrees m and n over an , having no common components, intersect in exactly m \times n points, counted with multiplicity. This equality holds in the complex \mathbb{P}^2, where points at ensure that even intersect, and multiplicity accounts for tangencies or higher-order coincidences at points. Named after the French mathematician (1730–1783), the theorem originated in his 1779 work Théorie générale des équations algébriques, where he provided proofs for specific cases involving resultants of polynomials, though without fully incorporating or multiplicities. Earlier ideas appeared in Isaac Newton's (1687), which observed that plane curves of degrees m and n intersect in up to m n points. The modern formulation, including multiplicity and , was refined in the 19th and 20th centuries, with algebraic treatments by figures like emphasizing . The theorem's significance lies in bridging algebraic and geometric perspectives, providing a precise count of solutions to systems of equations and enabling bounds on geometric configurations. In higher dimensions, generalizations extend to hypersurfaces in \mathbb{P}^n, where the of n hypersurfaces of degrees d_1, \dots, d_n has degree \prod d_i, again counted properly. Applications span , such as proving on conic sections, counting singular points on curves (at most \binom{d-1}{2} for degree d), and determining the genus of curves via the formula g = \frac{(d-1)(d-2)}{2}. It also plays a key role in and computational , underpinning algorithms for solving systems.

Historical Development

Early Origins

The roots of Bézout's theorem lie in , where foundational work on provided intuitive precursors to systematic counting of intersection points. In 's Elements (circa 300 BC), basic propositions on intersecting straight lines established the fundamental case where two lines meet at a single point, unless parallel, forming the simplest instance of intersection enumeration in . This approach influenced later studies by emphasizing geometric intersections without algebraic degrees. Apollonius of Perga advanced these ideas in his Conics (circa 200 BC), particularly in Book IV, which systematically examined the possible numbers of intersection points between conic sections—up to four for two conics—laying early groundwork for analyzing higher-degree curve interactions. These classical treatments focused on qualitative descriptions rather than general algebraic formulas, but they anticipated the need to quantify intersections across curve types. In the , algebraic methods began to formalize these concepts. , in Lemma 28 of volume 1 of his Principia Mathematica (1687), asserted that plane curves of degrees m and n intersect in exactly m \times n points, including complex or "imaginary" ones, though without explicit consideration of multiplicities and implicitly accounting for points at infinity through projective intuition. By the mid-18th century, Gabriel Cramer's Introduction à l'analyse des lignes courbes algébriques (1750) developed elimination theory for solving systems of polynomial equations, providing tools to determine common roots that later formed the basis for resultants in proofs of intersection theorems. These precursors set the stage for Étienne Bézout's rigorous algebraic refinements in the 1760s and 1770s.

Bézout's Contribution

Étienne Bézout advanced the algebraic foundations of in his 1764 memoir "Recherches sur les degrés des équations résultantes de l'évanouissement des inconnues," presented to the Académie Royale des Sciences. In this publication, he introduced resultants as a systematic method for eliminating variables from systems, enabling the determination of common roots and, by extension, the count of intersection points between algebraic curves defined by those equations. This approach provided a rigorous algebraic framework for quantifying intersections, marking a key step in the development of elimination theory, though it faced challenges with superfluous factors in non-generic cases. Bézout's contributions built on Isaac Newton's preliminary ideas on substitution-based elimination by generalizing them to systems involving multiple variables, addressing limitations in handling higher dimensions. His 1764 analysis offered an early proof of the intersection theorem for generic cases, but a more complete treatment appeared in his 1779 Théorie générale des équations algébriques, where he provided proofs for specific cases using resultants, without fully incorporating or multiplicities. These aspects were refined in the 19th and 20th centuries. The impact of Bézout's algebraic innovations extended into the 19th century, influencing projective geometers such as Jean-Victor Poncelet, whose synthetic methods in works like Traité des propriétés projectives des figures (1822) drew upon Bézout's resultant theory to explore curve intersections and polarity in projective spaces.

Statement

Plane Curves

Bézout's theorem in its classical form addresses the intersection properties of algebraic curves within the projective plane over an algebraically closed field. The projective plane \mathbb{P}^2_k over a field k is defined as the set of lines through the origin in the three-dimensional affine space k^3, where points are represented using homogeneous coordinates [x : y : z] with (x, y, z) \neq (0, 0, 0), and two triples are equivalent if one is a scalar multiple of the other by a nonzero element of k. An algebraic curve in \mathbb{P}^2_k is the zero set V(f) = \{ [x : y : z] \in \mathbb{P}^2_k \mid f(x, y, z) = 0 \}, where f \in k[x, y, z] is a nonzero homogeneous polynomial. The theorem applies to two such curves V(f) and V(g), where f is homogeneous of degree m and g is homogeneous of degree n, assuming k is algebraically closed and that V(f) and V(g) have no common irreducible components. Under these conditions, the curves intersect in exactly m n points in \mathbb{P}^2_k, counted with appropriate multiplicities and including any points at infinity. If the curves share common components, one factors out the of f and g, adjusts the degrees accordingly, and applies the theorem to the remaining factors to determine the intersection count. This result is expressed quantitatively through the , which sums the local multiplicities over all points in \mathbb{P}^2_k: \sum_{p \in \mathbb{P}^2_k} i_p(V(f), V(g)) = m n, where i_p(V(f), V(g)) denotes the multiplicity of the curves at the point p, a concept whose precise and properties are elaborated in subsequent sections.

In \mathbb{P}^n over an k, a is defined as the zero set V(f) of a single f \in k[x_0, \dots, x_n] of some degree m \geq 1. Hypersurfaces are subvarieties of 1 in \mathbb{P}^n. Bézout's theorem extends to the intersection of two such hypersurfaces V(f) = 0 of degree m and V(g) = 0 of degree n, where f and g form a (ensuring proper with no common components). Their is a of pure n-2 and degree m n, counted with multiplicity. This generalizes the classical case of plane curves in \mathbb{P}^2, where the consists of m n points. For multiple hypersurfaces, the theorem applies iteratively: the intersection of r hypersurfaces defined by a regular sequence of homogeneous polynomials of degrees d_1, \dots, d_r (with r \leq n) is equidimensional of dimension n - r, and for a complete intersection (when r = n), its degree is the product \prod_{i=1}^r d_i. Each successive pairwise intersection reduces the dimension by 1 while multiplying the degree by the next hypersurface degree. A specific instance occurs in \mathbb{P}^3, where two surfaces (hypersurfaces of degrees m and n) intersect in a of degree m n.

Affine Case

In the affine \mathbb{A}^2_k = k^2 over an k, algebraic are defined as the zero sets of non-constant f(x,y) \in k[x,y] and g(x,y) \in k[x,y], where the of a curve is the degree of its defining . These curves are obtained via dehomogenization of projective curves by setting the homogenizing variable z = 1. Bézout's theorem in the affine setting states that if two curves of degrees m and n have no common component, then they intersect in at most mn points in \mathbb{A}^2_k, counting multiplicities. This contrasts with the projective version, which guarantees exactly mn intersection points in \mathbb{P}^2_k. The affine count may be strictly less than mn because some intersections can occur at points at in the projective , which are excluded from the affine plane. For instance, two in \mathbb{A}^2_k, such as y = x and y = x + 1, intersect at exactly one in \mathbb{P}^2_k after homogenization, resulting in zero affine intersection points despite each having degree 1. Equality holds, yielding exactly mn affine intersection points counting multiplicities, if the curves have no common component and their leading homogeneous parts (the highest-degree terms) have no common zeros on the line \mathbb{P}^1_k. This condition ensures that the projective closures intersect properly without additional points or tangencies .

Intersection Multiplicity

Definition

In , the intersection multiplicity of two plane algebraic curves C and D at a point p provides a precise measure of their local intersection behavior, accounting for tangencies and singularities. For curves defined by polynomials f = 0 and g = 0 in the affine plane \mathbb{A}^2_k over an k, where p = (a, b) lies in the intersection V(f) \cap V(g), the multiplicity i_p(C, D) is given by the of the of the local ring at p by the generated by f and g: i_p(C, D) = \dim_k \left( \mathcal{O}_p / (f, g) \right), where \mathcal{O}_p is the local ring at p, obtained as the localization of k[x, y] at the maximal (x - a, y - b). This algebraic definition captures the extent to which the curves fail to intersect transversely at p; specifically, i_p(C, D) = 1 if the intersection is transverse (simple crossing with distinct s), while higher values indicate tangency or singular contact, such as i_p(C, D) = 2 for curves sharing a common line at p. Geometrically, the multiplicity quantifies the order of contact between the curves near p, reflecting how many "branches" or intersections coincide there, which is essential for applying Bézout's theorem to count total intersections properly. For instance, if one curve has a cusp or at p, the multiplicity adjusts the naive point count to preserve the theorem's product bound. In the projective setting, such as curves in \mathbb{P}^2_k, the multiplicity is defined locally in affine covering the : dehomogenize the homogeneous equations of the curves with respect to a chart containing p, compute the affine multiplicity as above, and ensure consistency across charts since the value is independent of the choice. An alternative computational approach uses when viewing the as elements of k, treating y as the variable: the order of the zero of the \operatorname{Res}_y(f, g) (a in x) at x = \alpha equals the sum of the multiplicities i_p(C, D) over all points p with x-coordinate \alpha. This method leverages elimination theory to detect the total multiplicity along the line x = \alpha without explicit localization. Overall, the sum of these local multiplicities over all points equals the product of the degrees of C and D, as stated in Bézout's theorem, ensuring the total is \deg C \cdot \deg D.

Key Properties

One key property of the intersection multiplicity i_p(C, D) at a point p is its invariance under changes of coordinates. Specifically, the multiplicity remains unchanged under affine transformations or projective transformations of the . This invariance extends to birational maps between curves, ensuring that local intersection behavior is preserved globally under such equivalences. Such stability under coordinate changes underscores the intrinsic nature of the multiplicity, often defined via the dimension of the quotient of the local ring at p by the ideal generated by the equations of C and D. Another fundamental property is additivity. When a curve C decomposes as the disjoint union C = C_1 \cup C_2, the intersection multiplicity satisfies i_p(C, D) = i_p(C_1, D) + i_p(C_2, D) at any point p. This follows from the more general additivity over factorizations of the defining polynomials, where for F = \prod F_i^{r_i} and G = \prod G_j^{s_j}, one has i_p(F \cap G) = \sum r_i s_j i_p(F_i \cap G_j). Additivity allows multiplicities to be computed componentwise, facilitating applications in resolving intersections. The Bézout identity provides a global constraint on intersection multiplicities. For two projective plane curves C and D of degrees m and n respectively, with no common irreducible components, the sum of the multiplicities over all points equals the product of the degrees: \sum_p i_p(C, D) = m n. This identity encapsulates the theorem's core assertion, counting intersections properly via multiplicities rather than geometric points alone. Intersection multiplicity also exhibits continuity with respect to perturbations of the equations. The value i_p(C, D) remains stable under small changes in the coefficients of the defining of C or D, such as replacing the equation of D with G + A F where A is a polynomial form of appropriate degree and F defines C. This property arises from the of dimensions in local rings and ensures that multiplicities persist under deformations, supporting enumerative applications of Bézout's theorem. Finally, for a line L passing through a multiple point p on a C, the intersection multiplicity i_p(C, L) equals the order of contact between L and C at p. This order measures the highest degree of vanishing of the restriction of of C along L, with higher orders corresponding to tangency or . Such equality highlights the multiplicity's role in quantifying local tangency conditions.

Examples

Two Lines

Bézout's theorem states that two projective plane curves of degrees m and n with no common components intersect in exactly mn points, counted with multiplicity. The simplest illustration arises when both curves are lines, each of degree 1, predicting a single intersection point. In the affine plane \mathbb{A}^2, a line is defined by a linear equation of the form ax + by + c = 0, where a, b, c \in \mathbb{R} (or more generally over an algebraically closed field like \mathbb{C}) and not both a and b are zero. For two such lines, L_1: ax + by + c = 0 and L_2: dx + ey + f = 0, their intersection is found by solving the corresponding linear system. If the lines are not parallel—meaning the determinant ae - bd \neq 0—there is a unique solution (x_0, y_0), yielding one intersection point. If parallel (ae = bd but the lines are distinct, so c/f \neq c'/f' wait no, more precisely if the coefficients are proportional but constants differ), the system has no solution, and the lines do not intersect in the affine plane. To resolve this in the broader geometric framework, consider the \mathbb{P}^2, where lines are homogenized to L_1: ax + by + cz = 0 and L_2: dx + ey + fz = 0, with points represented as [x : y : z]. Here, any two distinct lines intersect at exactly one point, as the homogeneous system always has a nontrivial solution up to scalar multiple. For non-parallel affine lines, the intersection remains the single affine point [x_0 : y_0 : 1]. For , the intersection occurs at a on the line z = 0: solving ax + by = 0 and dx + ey = 0 (since z=0) gives the direction perpendicular to the normal vectors, specifically [b : -a : 0] for L_1, which coincides for parallel L_2. Thus, Bézout's theorem holds with one intersection point counted in the projective closure. The intersection multiplicity at this point p is i_p(L_1, L_2) = 1 for distinct lines, as their tangents are transverse (not coincident). This follows from the definition of multiplicity as the dimension of the local ring quotient \dim_{\mathbb{C}} \mathcal{O}_{\mathbb{P}^2, p} / (F, G), which equals 1 for simple linear intersections without higher-order contact. In all cases, the total intersection number is $1 \times 1 = 1, verifying the theorem for this base case.

Line and Conic Section

In , Bézout's theorem applied to the intersection of a line and a conic section illustrates the general principle that two plane curves of degrees d_1 and d_2 intersect at d_1 d_2 points in the , counted with multiplicity. A line has 1, while a conic section, defined by a , has 2, so their intersections total 2 points. This holds over the complex numbers in \mathbb{P}^2, where points at are included to ensure the count is exact. To find the intersection points, parametrize the line and substitute into the conic equation. Consider a general line l(x,y) = 0 of degree 1 and a conic q(x,y) = 0 of degree 2. Parametrizing the line as (x,y) = (x_0 + t a, y_0 + t b) for direction (a,b) and point (x_0,y_0) on the line yields a quadratic equation in t upon substitution into q: q(x_0 + t a, y_0 + t b) = c_2 t^2 + c_1 t + c_0 = 0. The roots t_1, t_2 correspond to the intersection points, with multiplicities given by the root orders; distinct real roots indicate two distinct points, a double root indicates tangency with multiplicity 2 at that point, and complex roots may appear in the affine plane but are resolved projectively. In the , the total intersection multiplicity is always 2, accounting for points at . For instance, a tangent line to the conic intersects at a single affine point with multiplicity i_p = 2, satisfying the theorem without additional points. This contrasts with the affine plane, where some intersections may "escape" to , appearing as fewer than 2 points. The intersection multiplicity at a point p is defined algebraically as the dimension of the of the local ring at p by the ideals generated by the curve equations, ensuring the count includes tangency effects. A concrete affine example is the unit circle q(x,y) = x^2 + y^2 - 1 = 0 intersected with the line l(x,y) = y = 0. Substituting y = 0 gives x^2 - 1 = 0, with roots x = \pm 1, yielding two distinct points (\pm 1, 0), each with multiplicity 1. In projective coordinates [X:Y:Z], homogenizing to X^2 + Y^2 = Z^2 and the line Y = 0, the intersections remain these two points, with no additional at infinity since the line at infinity Z = 0 intersects the homogenized conic at [1:i:0] and [1:-i:0], but the specific line Y=0 does not pass through them. For a parabolic conic, consider q(x,y) = y - x^2 = 0 intersected with the line l(x,y) = y = 0, the at the . Substituting gives -x^2 = 0, a double root at x=0, so a single point (0,0) with multiplicity 2. In , homogenizing to Y Z - X^2 = 0 and Y = 0, the intersections are at [0:0:1] (affine , multiplicity 2) and confirmed total 2, with the point at [0:1:0] of the parabola not additionally intersected by this line. A line through the of the parabola, such as the vertical line x=0 (focus at (0, 1/4)), intersects at the finite point (0,0) with multiplicity 1 and at the point at [0:1:0] with multiplicity 1, illustrating how projective completion captures the full count.

Two Conic Sections

In , Bézout's theorem implies that two plane conic sections, defined by quadratic equations q_1(x,y) = 0 and q_2(x,y) = 0, intersect at exactly four points in the projective plane, counting multiplicities and points at infinity, provided they have no common component. For instance, consider the unit x^2 + y^2 = 1 and \frac{x^2}{4} + y^2 = 1. These curves can intersect at four real points, two real points and two points, or other configurations totaling four intersections when multiplicities are included. In the projective plane, conic sections may intersect at points at infinity. Two distinct circles, being special cases of conics, typically intersect at two finite points in the affine plane but also at the two circular points at infinity, [1 : i : 0] and [1 : -i : 0], to satisfy the four-point count. This resolves the apparent discrepancy where circles seem to intersect at only two points in the . Degenerate conics, such as a pair of lines represented by the equation xy = 0 (the coordinate axes), also fall under the theorem as degree-two curves. Such a degenerate conic intersects a non-degenerate conic at four points, counting multiplicities; for example, the lines x = 0 and y = 0 meet x^2 + y^2 = 1 at (0,1), (0,-1), (1,0), and (-1,0). When two conics are tangent at a point p, the intersection multiplicity i_p at that point is at least 2, which reduces the number of distinct intersection points while preserving the total count of four. This multiplicity arises from higher-order contact between the curves.

Proofs

Resultant Method

To prove Bézout's theorem using the resultant method, consider two plane algebraic curves defined by polynomials f(x, y) and g(x, y) of degrees m and n, respectively, over an . Dehomogenize the projective forms by setting the homogeneous coordinate Z = 1, yielding affine equations f(x, y) = 0 and g(x, y) = 0. Treat these as polynomials in y with coefficients in \mathbb{K}: f(x, y) = \sum_{i=0}^m a_i(x) y^i and g(x, y) = \sum_{j=0}^n b_j(x) y^j. The resultant \operatorname{Res}_y(f, g) is defined as the of the Sylvester , a square of size (m + n) \times (m + n) constructed from the coefficients a_i(x) and b_j(x). This vanishes if and only if f and g have a common in y for some value of x, corresponding to an point of the curves in the affine . As a polynomial in x, \operatorname{Res}_y(f, g) has degree at most mn. Assuming f and g have no common factors, the resultant is nonzero and exactly of mn. Its roots are precisely the x-coordinates of the points, each with multiplicity equal to the multiplicity i_p at the corresponding point p. Thus, the degree equation \deg \operatorname{Res}_y(f, g) = \sum i_p = mn establishes that the total number of intersections, counted with multiplicity, is mn. For the projective case, homogenize the polynomials to F(X, Y, Z) and G(X, Y, Z) of degrees m and n. The homogeneous resultant \operatorname{Res}_Y(F, G) is a homogeneous polynomial of degree mn in the remaining variables X, Z, vanishing at the projective points corresponding to intersections, again yielding the total multiplicity sum mn. This accounts for points at infinity. A key property linking the resultant to intersection counts is that, if f and g are coprime in \mathbb{K}, then \operatorname{Res}_y(f, g) = a_m^n \prod_{i=1}^m g(x, \alpha_i), where a_m is the leading coefficient of f in y and \alpha_i are the roots of f(x, y) = 0 in y (treating x as fixed). The zeros of this product occur where g(x, \alpha_i) = 0, precisely at the x-coordinates of intersections.

U-Resultant Approach

The U-resultant approach provides an alternative proof of Bézout's theorem for the intersection of plane curves using a resultant construction that incorporates auxiliary , offering a symmetric formulation suited to . For two homogeneous polynomials F, G \in k[X, Y, Z] of degrees m and n over an k, introduce auxiliary variables u_0, u_1, u_2 and consider the bihomogeneous system u_1 F - u_0 G = 0 in the variables Y, Z (treating X as parameter, or symmetrically). More generally, the U-resultant is the with respect to Y, Z of the polynomials u_0 G(X, Y, Z) - u_1 F(X, Y, Z) and related terms to enforce the u_0 X + u_1 Y + u_2 Z = 0, but in practice for two equations, it reduces to a formulation equivalent to the classical resultant up to constants. This U-resultant is a bihomogeneous of mn in the coefficients of F and G, and also linear in the u_i. Its vanishing corresponds to the curves having a common point in \mathbb{P}^2, with the multiplicity of intersection points given by the multiplicity of the corresponding linear in the u_i. Thus, since the is mn, the curves intersect in exactly mn points in the , counted with multiplicity. The underlying proof relies on the fact that the U-resultant factors into linear terms over the points, each corresponding to the projective coordinates [u_0 : u_1 : u_2] of the point, with multiplicity reflecting the local order. This provides an explicit whose roots encode the intersection locations without choosing coordinates. This approach has advantages over the Sylvester resultant method, particularly its inherent for homogeneous polynomials, which simplifies computations in , and its explicit construction for low-degree cases like conics. For instance, when F and G are quadratic forms (quadrics, m = n = 2), the U-resultant can be computed via a resultant matrix of size related to the degrees, and the four intersection points can be recovered by solving for the ratios in the auxiliary variables or analyzing the to find the points.

Ideal Degree Method

In the context of , the ideal degree method offers a commutative algebraic proof of Bézout's theorem by analyzing the degree of the ideal generated by the equations of two curves in the . Consider the \mathbb{P}^2_k over an k, with homogeneous coordinate ring R = k[x, y, z]. Let f \in R_m and g \in R_n be homogeneous polynomials of s m and n, respectively, generating the graded I = (f, g). Assume I defines a , meaning f and g form a (so g is not a zero-divisor in R/(f)) and share no common irreducible components, ensuring \height(I) = 2. The quotient R/I is then a finitely generated graded R-module of Krull dimension 1, whose Proj corresponds to the 0-dimensional scheme V(f, g) \subset \mathbb{P}^2_k consisting of the intersection points (counted with multiplicity). The degree of this scheme, which equals the total intersection multiplicity, is the leading coefficient (normalized by the factorial) of the Hilbert polynomial P_{R/I}(d) of R/I. For a complete intersection of this type, this degree is precisely m n. This degree can be computed using the Hilbert series of R/I, derived from the Koszul complex resolution $0 \to R(-m-n) \xrightarrow{(g, -f)} R(-m) \oplus R(-n) \xrightarrow{(f, g)} R \to R/I \to 0, which is exact under the complete intersection assumption. The Hilbert series is thus HS_{R/I}(t) = \frac{1 - t^m - t^n + t^{m+n}}{(1 - t)^3}, since HS_R(t) = 1/(1 - t)^3. The associated Hilbert polynomial P_{R/I}(d) is linear, P_{R/I}(d) = m n \, d + \chi(R/I), where the slope m n gives the degree of V(f, g). Equivalently, the degree multiplies iteratively: \deg(R/(f)) = m, and adjoining g (a non-zero-divisor modulo (f)) yields \deg(R/(f, g)) = m n. The multiplicities at individual intersection points arise from the primary decomposition of I into primary ideals associated to maximal ideals corresponding to those points; the sum of the lengths of these primary components equals the total degree m n. More explicitly, the Hilbert function h_{R/I}(d) = \dim_k (R/I)_d is h_{R/I}(d) = \binom{d+2}{2} - \binom{d-m+2}{2} - \binom{d-n+2}{2} + \binom{d-m-n+2}{2} (with binomial coefficients zero if the upper index is negative), which agrees with the Hilbert polynomial for large d and whose leading behavior confirms the degree m n, via properties of resolutions and Macaulay's bounds on Hilbert functions. This approach extends to more general schemes, where the class [V(f, g)] = m n [\mathrm{pt}] in the Chow ring of \mathbb{P}^2_k, capturing the intersection product without relying on explicit coordinates.