Bézouts

Bézouts refers to several well-known results named after Étienne Bézout, an 18th‑century French mathematician recognized for his work in elimination theory and algebraic geometry. The most frequently cited are Bézout's identity (often called Bézout's lemma) and Bézout's theorem.

Bézout's identity states that for integers a and b with greatest common divisor d, there exist integers x and y such that ax + by = d. Consequently, d is the smallest positive integer that can be expressed as a linear combination of a and b. The Extended Euclidean algorithm computes such coefficients x and y and is widely used to solve linear Diophantine equations.

Bézout's theorem is a fundamental result in projective algebraic geometry. It asserts that, over an algebraically closed field, two plane curves of degrees m and n that have no common component intersect in exactly mn points, counted with multiplicity. This global intersection count is valid after homogenization to projective space, and deviations occur if the curves share a component or the base field is not algebraically closed. The theorem underpins elimination theory and the construction of resultants; it also has polynomial-analogues describing the number of common zeros in one variable.

Bézout coefficients refer to the integer pair x and y in Bézout's identity. While not unique, various algorithms (notably the Extended Euclidean algorithm) produce one such pair for given a and b.

Historically, Bézout’s work laid foundations for methods to eliminate variables and analyze polynomial equations, influencing both number theory and algebraic geometry.