A determinant that vanishes exactly when two polynomials share a root
The resultant of two polynomials is the determinant of their Sylvester matrix; it vanishes precisely when they have a common root. Computing it as a literal determinant is far too expensive, so it is obtained from a polynomial remainder sequence whose coefficient growth is controlled by subresultant theory. The discriminant — the resultant of a polynomial with its own derivative — detects repeated roots, and for a number field it controls which primes ramify.
Learning objectives
- Define the resultant by the Sylvester matrix and by the product over roots.
- Explain how the subresultant PRS controls coefficient growth.
- Use resultants to eliminate a variable from a system.
- Relate the discriminant to repeated roots and to ramification.
- Recognise where resultants become impractical.
Section 01The resultant
For A of degree m and B of degree n, the Sylvester matrix is the (m+n)-square matrix of their shifted coefficient rows. Its determinant is the resultant, and it satisfies
The product formula runs over the roots and shows immediately that the resultant vanishes exactly when a root is shared. The determinant formula shows that it is a polynomial in the coefficients with integer coefficients — so it can be computed exactly, without ever approximating a root.
This is the point of the resultant. Given two equations in x and y, the resultant with respect to y is a single polynomial in x alone whose roots include every x-coordinate of a common solution. The variable is eliminated algebraically — no numerical root-finding is involved.
Section 02The subresultant remainder sequence
- Set R0 ← A, R1 ← B; set g ← 1, h ← 1.
- Set δ ← deg Ri−1 − deg Ri.
- Compute the pseudo-remainder R of Ri−1 by Ri.
- Set Ri+1 ← R / (g hδ), with sign adjustment. This division is exact — that is the content of subresultant theory.
- Update g ← lc(Ri) and h ← h1−δ gδ.
- Repeat until the remainder is zero; the resultant is recovered from the last non-zero term and the accumulated factors.
The subresultant sequence delivers the GCD and the resultant together. If the last non-zero remainder is a non-zero constant the polynomials are coprime and that constant determines the resultant; otherwise the GCD is non-trivial and the resultant is zero.
Section 03Discriminants
The discriminant of A is essentially the resultant of A with its derivative:
It vanishes exactly when A has a repeated root. In number theory the discriminant carries far more information than that single test.
Squarefree testing
A non-zero discriminant certifies distinct roots — the precondition for most factorisation algorithms.
Ramification
A rational prime ramifies in a number field exactly when it divides the field discriminant. This is the bridge from polynomial data to arithmetic structure.
Index bounding
disc(f) = [ℤK : ℤ[θ]]2 · dK, so the square factors of the polynomial discriminant bound the index of the equation order in the maximal order.
Quadratic fields
The field discriminant determines the field completely and indexes the class number tables.
Elliptic curves
A non-zero curve discriminant is the condition for non-singularity; its prime factors are the primes of bad reduction.
Mass formulae
Discriminant bounds constrain the possible number fields of a given degree and signature.
Section 04Practical limits
Resultants grow quickly. The resultant of two polynomials of degree n with d-digit coefficients has degree up to 2n in the remaining variable and coefficients of roughly 2nd digits. Eliminating repeatedly through a system of several variables compounds this at every step.
Successive elimination of three or more variables produces objects of unmanageable size, and introduces spurious solutions that must afterwards be filtered out. Gröbner bases or triangular decompositions are the appropriate tools for multivariate systems; resultants remain excellent for two polynomials in two variables.
ReferenceFrequently asked questions
Is the Sylvester determinant ever computed directly?
Only for very small degrees or as a correctness check. The matrix has dimension m + n and its entries are the coefficients, so exact expansion is far more expensive than the remainder sequence.
What does a zero resultant tell me over Z?
That the polynomials have a common factor of positive degree over ℚ, and hence over ℤ after removing content. It does not by itself identify the factor — the GCD computation, which the same remainder sequence provides, does that.
Why does the polynomial discriminant differ from the field discriminant?
Because the equation order ℤ[θ] may be a proper subring of the ring of integers. The two differ by the square of the index, which is why determining the maximal order requires factoring the square part of the polynomial discriminant.
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