The Resultant

Take two polynomials, say $f(x) = x^2 - 5x + 6$ and $g(x) = x^2 - 8x + 15$. Do they share a root? You could factor both: $f(x) = (x-2)(x-3)$ and $g(x) = (x-3)(x-5)$, and spot the common factor $(x-3)$. That works fine for small, tidy examples.

But factoring is not always an option. What if the coefficients are symbolic, or the polynomials have degree 20? You need a way to answer "do they share a root?" using nothing but the coefficients themselves — arithmetic, not guesswork.

That is exactly what the resultant does. Build a certain matrix from the coefficients of $f$ and $g$ — the Sylvester matrix — take its determinant, and the result is a single number: the resultant $\operatorname{Res}(f,g)$. It is zero if and only if $f$ and $g$ share a root (over the complex numbers). No factoring, no root-finding — just arithmetic on the coefficients.

Fix $g(x) = x^2 - 5x + 6 = (x-2)(x-3)$. Now take a family $f(x) = x^2 + bx + 2b$ and slide $b$. For most values of $b$, $f$ and $g$ have no root in common. But the resultant $\operatorname{Res}(f,g)$, computed straight from the coefficients as a $4\times4$ determinant, tells you exactly when a shared root appears — without ever solving for the roots themselves.

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Drag the slider slowly. Watch the resultant curve and the two parabolas at the same time: every time the resultant crosses zero, the two curves are touching at a common point. The determinant "feels" the tangency algebraically, before you ever look at the picture.

The construction is completely mechanical. For $f(x) = a_0 x^m + a_1 x^{m-1} + \dots + a_m$ and $g(x) = b_0 x^n + b_1 x^{n-1} + \dots + b_n$, the Sylvester matrix is an $(m+n)\times(m+n)$ matrix built from shifted copies of the coefficients of $f$ (n rows) stacked on shifted copies of the coefficients of $g$ (m rows). Its determinant is the resultant:

$$\operatorname{Res}(f,g) = \det S(f,g)$$

• Why it works: $\operatorname{Res}(f,g) = 0$ exactly when $f$ and $g$ have a nontrivial common factor — equivalently, a common root in the complex numbers (or a shared root "at infinity" if leading coefficients vanish). This falls out of linear algebra: the rows of the Sylvester matrix encode the coefficients of $x^i f(x)$ and $x^j g(x)$, and a common root forces a linear dependency among them, which is exactly what a zero determinant means.
• It is a polynomial in the coefficients. $\operatorname{Res}(f,g)$ is itself a polynomial expression in the $a_i$ and $b_j$ — no radicals, no case-splitting. That makes it exact and symbolic, unlike numerically hunting for roots.
• It generalizes. Setting $g = f'$ (the derivative) gives the discriminant, which detects repeated roots of a single polynomial — the same determinant machine, one polynomial away from the resultant.
• Cost: computing an $(m+n)\times(m+n)$ determinant directly costs $O((m+n)^3)$ with standard methods, though faster resultant-specific algorithms exist. It is polynomial-time, symbolic, and exact — a very different profile from problems like P vs NP, where no such efficient exact recipe is known.

"Eliminate a variable between two equations" sounds narrow, but it is one of the most-needed operations in computational mathematics:

• Elimination theory: given two polynomial equations in $x$ and $y$, treating them as polynomials in $x$ with coefficients in $y$ and taking the resultant produces a single equation in $y$ alone — describing exactly where two curves intersect.
• Computer-aided design and robotics: solving for the configurations where two algebraic curves or surfaces meet (collision detection, linkage kinematics) routinely reduces to a resultant computation.
• Computer algebra systems: resultants underlie variable-elimination routines that complement Gröbner bases for solving systems of polynomial equations exactly.
• Number theory and cryptanalysis: resultants help detect common factors between polynomials over finite fields, relevant to constructing and attacking certain cryptographic schemes.

Any time "do these two algebraic conditions simultaneously hold" needs a symbolic, no-approximation answer, the resultant is often the tool that gets there fastest.

The resultant turns a question that looks like it needs solving — "do these two polynomials share a root?" — into a question about a determinant, answerable by pure arithmetic on the coefficients. Zero means yes, they touch; nonzero means no, they never do.

It is a small piece of a bigger idea: sometimes the fastest way to answer a hard-looking question is not to solve the problem in front of you, but to build the right algebraic object — a matrix, a determinant, an invariant — that already knows the answer. The same spirit shows up throughout Gröbner bases and the rest of computational algebraic geometry.