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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|02SJG1HC32]].\nLet $C$ and $D$ be projective [[plane curves|EWS4WFTA7W]] in the [[projective plane|ZT9A6JK8QF]] $\\mathbb{P}^2_k$.\nSuppose they are defined by homogeneous polynomials of degrees $d$ and $e$ with no common factor of positive degree.\nThen $C \\cap D$ is finite, and\n$$\\sum_{P \\in C \\cap D} i_P(C, D) = d e,$$\nwhere $i_P(C, D)$ is the [[intersection multiplicity|GPQF9GK4T9]] of $C$ and $D$ at $P$.\n\n## Remarks\n\nOne proof uses the resultant of the two polynomials with respect to one variable, after a change of coordinates.\nIt is a homogeneous polynomial of degree $de$ in the other two variables, and its roots, with multiplicity, correspond to the intersection points.\nIn [[intersection theory|4HV4WTAX9P]] the proof is one line: $[C] = d h$ and $[D] = e h$ in $CH^\\bullet(\\mathbb{P}^2) = \\Z[h]/(h^3)$, so $[C] \\cdot [D] = de \\, h^2$, and $h^2$ is the class of a point.\nExample: a line and a smooth conic meet in two points, or in one point of multiplicity $2$ when the line is tangent.\nEach hypothesis is needed.\nOver $\\R$, the circle $x^2 + y^2 = z^2$ and the line $x = 2z$ have no common real point, but they meet in two complex points.\nIn the affine plane, two parallel lines do not meet; in $\\mathbb{P}^2$ they meet at a point at infinity.\nWithout multiplicities, a tangent line meets a conic in only one point.\nIf the polynomials have a common factor, the curves share a component and meet in infinitely many points.\nMore generally, $n$ hypersurfaces of degrees $d_1, \\ldots, d_n$ in $\\mathbb{P}^n_k$ with finite intersection meet in $d_1 \\cdots d_n$ points counted with multiplicity.\n\n## Explanations\n\n### L1 human\n\nTwo curves in the plane, given by equations of degrees $d$ and $e$ with no common factor, meet in exactly $de$ points.\nThis is true once three adjustments are made: complex points are allowed, points at infinity are included, and tangency points are counted several times.\nFor example, two conics without a common component meet in four points, counted properly.\n\n### L2 human\n\nBézout's theorem is the first result of [[intersection theory|4HV4WTAX9P]], and the model for the general principle that intersection numbers depend only on the classes of the cycles.\nIt gives the bound of at most $de$ common points for any two plane curves without common component, over any field.\nTogether with the genus formula it is the basic tool of the classical theory of plane curves.\n",
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