A structured matrix associated to two univariate polynomials whose entries are bilinear forms in the polynomial coefficients; its determinant (up to a scalar) equals the resultant and its kernel encodes common roots and gcd information.

Principle

Construct the Bezoutian bilinear form B(f,g)(u,v) = (f(u)g(v) - f(v)g(u))/(u-v) and represent it in monomial bases to obtain the Bezout matrix; algebraic properties of f and g (common roots, multiplicities) correspond to rank and kernel properties of that matrix.

Demonstration

For f(x)=a_0 + a_1 x + ... + a_m x^m and g(x)=b_0 + b_1 x + ... + b_n x^n the Bezout matrix of size max(m,n) has entries given by coefficients appearing in the symmetric expansion of the Bezoutian. Its determinant vanishes exactly when f and g have a common root, producing the resultant.

Misapplication

Using the Bezout matrix without accounting for numerical conditioning can lead to unstable root computations; conflating the Bezout matrix with the Sylvester matrix or using it naively for multivariate elimination without adaptation yields incorrect conclusions.

Consequence

Proper use gives a compact algebraic test for common roots, a route to compute resultants, to detect multiplicities and to form structured linearizations for numerical eigenvalue methods that isolate common roots.

Reversal

The Sylvester matrix is a different elimination matrix built from coefficient convolution whose determinant is also the resultant; reversing choice clarifies trade-offs: Sylvester is often sparser, Bezout encodes a symmetric bilinear form with different numerical and algebraic properties.

Boundary

Definition is standard for univariate polynomials over a field or principal ideal domain; extensions to multivariate polynomials require elimination or projection and are not direct. Scaling, base change, and leading-zero coefficient cases must be handled to avoid rank-deficiency artefacts.

Semantic Tension

Tension exists between using Bezout matrices as algebraic exact-result tools and their behavior in numerical settings where rounding and conditioning matter; also between different resultant matrices (Sylvester, Dixon, Bézoutian) which represent the same resultant differently.

Synthesis

The Bezout matrix is the matrix realization of the Bezoutian bilinear form built from two polynomials; its determinant yields the resultant and its linear algebra (rank, nullspace) precisely reflects common roots and multiplicities, providing both symbolic elimination and structured numerical approaches.