zeezout

Zeezout is an alternative rendering of Bézout, a surname attached to several results in algebra and number theory named after the French mathematician Étienne Bézout (1730–1783). The term is associated most often with results concerning greatest common divisors and intersections of algebraic objects.

In number theory, Bezout's identity states that for integers a and b, there exist integers x and y such that ax + by = d, where d is the greatest common divisor of a and b. Consequently gcd(a,b) is the smallest positive integer expressible as a linear combination of a and b. This result underpins the extended Euclidean algorithm, which simultaneously computes gcd(a,b) and such coefficients x and y.

In algebra, for polynomials A(x) and B(x) over a field F, there exist polynomials X(x) and Y(x) such that AX + BY = gcd(A,B). This expresses the gcd of the polynomials as a linear combination and plays a key role in polynomial division and elimination theory.

Bezout's theorem in algebraic geometry states that two projective plane curves of degrees m and n with no common component intersect in exactly mn points in the complex projective plane, counted with multiplicity. This foundational result ties together degree, intersection theory, and the geometry of curves, and has many generalizations to higher dimensions and other fields.

Related notions include the resultant, a determinant that vanishes when two polynomials share a root, and the Bezout matrix, which encodes information about common roots and gcds. Étienne Bézout's work laid important groundwork for elimination theory and the algebraic study of system of equations.