Function Repository Resource:

# ExtendedGroebnerBasis

Compute a Groebner basis and a conversion matrix from the input polynomials to the basis

Contributed by: Daniel Lichtblau
 ResourceFunction["ExtendedGroebnerBasis"][polys,vars] gives the Groebner basis 𝒢 for the polynomial list polys along with a polynomial conversion matrix ℳ satisfying 𝒢=ℳ·polys.

## Details and Options

ResourceFunction["ExtendedGroebnerBasis"] takes the same optional arguments and options as GroebnerBasis.

## Examples

### Basic Examples (5)

Create a set of polynomials:

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First compute its Groebner basis:

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Compute the extended Groebner basis:

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Check that the bases agree up to a constant factor:

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Check the conversion matrix identity:

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### Scope (3)

Here is a larger example:

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Check that the bases agree up to constant factors (they are not identical in this instance):

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Check the conversion matrix identity:

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### Properties and Relations (7)

The Groebner basis of a pair of univariate polynomials is their greatest common divisor, and similarly the extended Groebner basis gives the extended GCD. Create a pair of polynomials with a known common divisor poly1:

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Compute the extended GCD:

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Check that the single polynomial in the Groebner basis is a GCD (a constant multiple of poly1):

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Check the Bezout relation for the extended GCD:

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Compare to PolynomialExtendedGCD:

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Again check the Bezout relation:

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Now check that these two computations agree up to a constant factor:

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### Possible Issues (1)

ExtendedGroebnerBasis can be much slower that GroebnerBasis for the same input:

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## Version History

• 1.0.0 – 27 September 2019