Bézout's theorem in the context of Degree of a polynomial

⭐ Core Definition: Bézout's theorem

In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of $n$ polynomials in $n$ indeterminates. In its original form the theorem states that in general the number of common zeros equals the product of the degrees of the polynomials. It is named after Étienne Bézout.

In some elementary texts, Bézout's theorem refers only to the case of two variables, and asserts that, if two plane algebraic curves of degrees $d_{1}$ and $d_{2}$ have no component in common, they have $d_{1}d_{2}$ intersection points, counted with their multiplicity, and including points at infinity and points with complex coordinates.

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⭐ Core Definition: Bézout's theorem
Bézout's theorem in the context of Degree (algebraic geometry)
Bézout's theorem in the context of Self-intersection

Bézout's theorem in the context of Degree (algebraic geometry)

In mathematics, the degree of an affine or projective variety of dimension $n$ is the number of intersection points of the varietywith $n$ hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of Bézout's theorem. (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem.)

The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space.

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Bézout's theorem in the context of Self-intersection

In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form.

There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov–Witten theory and the extension of intersection theory from schemes to stacks.

View the full Wikipedia page for Self-intersection

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