The structured Toeplitz-like matrix built from coefficient vectors of two univariate polynomials whose determinant equals their resultant. Rows consist of shifted copies of the polynomials' coefficient sequences so that linear dependence of columns corresponds to a common root.

Principle

Translate polynomial multiplication and coefficient-shifting into a linear algebra object: align shifted coefficient vectors of f and g into one rectangular matrix so that a nontrivial nullvector corresponds to a common polynomial multiple, and the determinant of the square Sylvester matrix is the elimination criterion.

Demonstration

Given f of degree m and g of degree n, form a matrix with n rows of shifted f-coefficients and m rows of shifted g-coefficients to obtain an (m+n)×(m+n) Sylvester matrix. Its determinant vanishes exactly when f and g have a common root; rank drops reveal the degree of the gcd.

Misapplication

Constructing the Sylvester matrix with incorrect shift order or ignoring leading coefficient scaling leads to wrong determinant values; using it blindly for multivariate systems without passing to Macaulay or resultant generalizations misrepresents solvability.

Consequence

The Sylvester matrix converts a polynomial common-root problem into rank and determinant problems: determinant gives resultant, rank deficiency gives gcd degree, and nullspace vectors produce annihilating polynomials useful for factor extraction.

Reversal

Contrast with the Bezout or Hankel constructions: while Sylvester arranges coefficient shifts, Bezout matrices encode bilinear forms of evaluation; reversing perspective means using dual bilinear encodings rather than shifted rows to capture common factors.

Boundary

Defined classically for two univariate polynomials; for more polynomials or multivariate cases one uses Macaulay or sparse resultant matrices. Over rings with zero divisors the linear-algebraic implications of determinant vanishing require extra hypotheses.

Semantic Tension

Tension exists between Sylvester and Bezout matrices: both yield resultants but differ in numerical conditioning, symmetry, and how multiplicity information appears; choice depends on the subsequent computational goals (rank vs bilinear structure).

Synthesis

The Sylvester matrix is an explicit linear-algebra encoding of shifted polynomial coefficients whose determinant is the resultant: by converting elimination to matrix rank and determinant computations it connects polynomial solvability with classical matrix analysis.