Polynomial Arithmetic and GCD
Resultants and Discriminants
The resultant as a criterion for common roots, the discriminant as a test for repeated roots, and how both are computed in practice.
Engineering / MathematicsPolynomial Arithmetic and GCD2 min readKV-MATH-0561
The resultant detects whether two polynomials share a root without computing any roots. The discriminant is the resultant of a polynomial with its derivative, detecting repeated roots. Both are central to number field computation.
The resultant
The resultant of two polynomials is the determinant of their Sylvester matrix, and it vanishes exactly when they have a common root in an algebraic closure.
Computing it
| Method | Approach | Suits |
|---|---|---|
| Sylvester determinant | Direct determinant of a matrix of size the sum of the degrees | Small degrees; conceptually clear |
| Sub-resultant sequence | Read off the last non-zero remainder | The standard practical method |
| Modular with CRT | Compute modulo primes and reconstruct | Large coefficients |
| Evaluation-interpolation | For multivariate resultants | Elimination problems |
The discriminant
- Zero discriminant
- The polynomial has a repeated root, hence is not squarefree.
- Polynomial discriminant of a number field generator
- Differs from the field discriminant by the square of the index — see discriminants and integral bases.
- Sign over the reals
- Encodes the number of real roots for small degrees, and the signature of the field for a defining polynomial.
Elimination
Resultants eliminate variables: taking the resultant of two multivariate polynomials with respect to one variable produces a polynomial in the others that vanishes wherever both originals do.
Uses in this collection
- Computing discriminants of defining polynomials, the first step toward the maximal order.
- Constructing resolvent polynomials in Galois group computation.
- Computing minimal polynomials of sums and products of algebraic numbers.
- Testing squarefreeness before factorisation.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 3.3.2. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.