The Subresultant Algorithm, Resultants and Discriminants

Executive summary

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

Res(A, B) = lc(A)ⁿ ∏_i B(α_i) = (−1)^mn lc(B)^m ∏_j A(β_j)

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.

Elimination without solving

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

AlgorithmSubresultant PRSin: A, B over a UFD → out: the PRS, the GCD and the resultant

Set R₀ ← A, R₁ ← B; set g ← 1, h ← 1.
Set δ ← deg R_i−1 − deg R_i.
Compute the pseudo-remainder R of R_i−1 by R_i.
Set R_i+1 ← R / (g h^δ), with sign adjustment. This division is exact — that is the content of subresultant theory.
Update g ← lc(R_i) and h ← h^1−δ g^δ.
Repeat until the remainder is zero; the resultant is recovered from the last non-zero term and the accumulated factors.

Two results from one computation

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:

disc(A) = (−1)^m(m−1)/2 Res(A, A′) / lc(A)

It vanishes exactly when A has a repeated root. In number theory the discriminant carries far more information than that single test.

Role

Squarefree testing

A non-zero discriminant certifies distinct roots — the precondition for most factorisation algorithms.

Role

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.

Role

Index bounding

disc(f) = [ℤ_K : ℤ[θ]]² · d_K, so the square factors of the polynomial discriminant bound the index of the equation order in the maximal order.

Role

Quadratic fields

The field discriminant determines the field completely and indexes the class number tables.

Role

Elliptic curves

A non-zero curve discriminant is the condition for non-singularity; its prime factors are the primes of bad reduction.

Role

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.

Do not build an elimination pipeline on iterated resultants

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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This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review The Subresultant Algorithm, Resultants and Discriminants. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat The Subresultant Algorithm, Resultants and Discriminants as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—subresultant, resultant, section, discriminants, polynomial—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying The Subresultant Algorithm, Resultants and Discriminants?