Polynomial Resultant

Concept Overview

Given two monic quadratic polynomials f(x) = x2 + ax + b and g(x) = x2 + cx + d, asking whether they share a complex root x0 is equivalent to asking whether there exist non-zero linear multipliers p(x) = p1x + p0 and q(x) = q1x + q0 such that p(x)f(x) + q(x)g(x) = 0.

Expanding this polynomial identity in the monomial basis {x^3, x^2, x, 1} produces a homogeneous 4×4 linear system whose coefficient matrix is the Sylvester matrix S(f, g).

Mathematical Formulation

For f(x) = x2 + ax + b and g(x) = x2 + cx + d, the 4×4 Sylvester matrix has rows [1, a, b, 0], [0, 1, a, b], [1, c, d, 0], and [0, 1, c, d]. Its determinant is the resultant Res(f, g) = det S(f, g).

By subtracting the two quadratics, any common root must satisfy g(x) - f(x) = (c - a)x + (d - b) = ux + v = 0, where u = c - a and v = d - b. Eliminating x yields the closed-form resultant formula Res(f, g) = v2 - auv + bu2.

Worked Example

Let us calculate the resultant for two distinct monic quadratics:
f(x) = x² + x - 2
g(x) = x² - 2x + 2

Here, the coefficients are a = 1, b = -2, c = -2, d = 2. We form the 4×4 Sylvester matrix:

[
  [1,  1, -2,  0],
  [0,  1,  1, -2],
  [1, -2,  2,  0],
  [0,  1, -2,  2]
]

We can calculate the determinant of this matrix, which is the resultant, by using the elimination relation:

1. Find u = c - a = -2 - 1 = -3.
2. Find v = d - b = 2 - (-2) = 4.
3. Calculate the resultant: Resultant = v² - auv + bu².
4. Resultant = (4)² - (1)(-3)(4) + (-2)(-3)² = 16 + 12 - 18 = 10.

Since the resultant is 10 (nonzero), the two polynomials do not share a common complex root.

Experiment

You can test the resultant interactively in the Polynomial Resultant module. Try the following steps:

1. Set a to 1, b to 1, c to 1, and d to 1 to represent identical polynomials.
2. Observe that the Resultant becomes exactly 0.
3. The Common complex root indicator updates to true, as identical polynomials share all roots.
4. Now change a to 0, b to -1, c to -2, and d to 1.
5. The Resultant again evaluates to 0, showing that f(x) = x² - 1 and g(x) = x² - 2x + 1 share the real root x = 1.

Limitations

This module is specifically designed for monic quadratic polynomials (where the leading coefficient is 1). The formula and 4×4 Sylvester matrix shown are not generalizable to polynomials of higher or lower degrees without altering the matrix dimensions and elements. Furthermore, a resultant of zero indicates that the polynomials share at least one complex root, but it does not specify whether the shared root is real or non-real complex.

Common Pitfalls

Do not confuse the resultant Res(f, g) of two polynomials with the discriminant of a single polynomial. A polynomial such as f(x) = x2 has a repeated root at x = 0 (zero discriminant), and g(x) = (x - 1)2 has a repeated root at x = 1, yet their resultant is Res(f, g) = 1 ≠ 0 because they do not share a root with each other.

References

The following primary references provide algorithms for computing resultants over general rings using Sylvester matrices.

• SymPy Documentation: Polynomials Internals — Provides definitions and algorithms for polynomial resultants and Sylvester matrices.