PolynomialTraits_d::Resultant

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PolynomialTraits_d::Resultant

Definition

This AdaptableBinaryFunction computes the resultant of two polynomials f and g of type PolynomialTraits_d::Polynomial_d with respect to a certain variable.

Note that this functor operates on the polynomial in the univariate view, that is, the polynomial is considered as a univariate polynomial in one specific variable.

Let f and g be two univariate polynomials over some commutative ring A, where

f = f_mx^m + … + f₀

and

g = g_nxⁿ + … + g₀.

The resultant of f and g is defined as the determinant of the Sylvester matrix:

Note that this is a (n+m) × (n+m) matrix as there are n rows for f and m rows that are used for g. The blank spaces are supposed to be filled with zeros.

Let L be the algebraic closure of A, and write f and g as

f := f_m

∏

i=1

(x-α_i)

, α_i ∈ L

and

g := g_n

∏

j=1

(x-β_j)

, β_i ∈ L,

then the resultant of f and g is (up to leading coefficients) the product of all pairwise differences of the roots of f and g, namely

res(f,g) = f_mⁿ g_n^m

∏

i=1

∏

j=1

(α_i-β_j)

In particular, res(f,g) ≠ 0 iff f and g have a common factor with a positive degree in X.

There are various ways to compute the resultant. Naive options are the computation of the resultant as the determinant of the Sylvester Matrix or the Bezout Matrix as well as the so called subresultant algorithm, which is a variant of the Euclidean Algorithm. More sophisticated methods may use modular arithmetic and interpolation. For more information we refer to, e.g., [vzGG99].

Refines

AdaptableBinaryFunction
CopyConstructible
DefaultConstructible

Types

typedef PolynomialTraits_d::Coefficient_type
	result_type;
typedef PolynomialTraits_d::Polynomial_d
	first_argument_type;
typedef PolynomialTraits_d::Polynomial_d
	second_argument_type;

Operations

result_type	fo ( first_argument_type f , second_argument_type g )
		Computes the resultant of f and g, with respect to the outermost variable.