Bezouts

Bezouts refers to several concepts named after the French mathematician Étienne Bézout (1730–1783). The most notable are Bézout's identity, Bézout's theorem, and the Bézout matrix (also called the Bezoutian).

Bézout's identity: If a and b are integers with greatest common divisor d, then there exist integers x and y such that ax + by = d. In particular, d divides any linear combination of a and b. The set of all such combinations forms the ideal generated by a and b in the ring of integers. This result underpins many arguments in elementary number theory and continues to be used in algorithms for computing gcds and modular inverses.

Bézout's theorem: In the projective plane, two algebraic curves of degrees m and n with no common component intersect in mn points, counting multiplicities and points at infinity; this holds over algebraically closed fields. The theorem provides a fundamental link between the degrees of curves and their intersection behavior, and it serves as a cornerstone of algebraic geometry. Generalizations extend to higher dimensions and consider multiplicities, residual components, and singularities.

Bézout matrix (the Bezoutian): Given two polynomials p and q, the Bézout matrix is a symmetric matrix constructed from their coefficients such that its determinant equals the resultant of p and q, up to a scalar factor. The determinant vanishes precisely when p and q have a common root, making the matrix a tool in resultant theory and polynomial system solving.

The name also appears in related elimination and resultant theories and reflects Bézout’s contributions to polynomial algebra and algebraic geometry.