An algorithm that constructs the interpolation ideal and a Gröbner basis from a finite set of points by incrementally building linear systems for monomial evaluations, using linear algebra to determine relations among monomials and produce bases of the vanishing ideal.

Principle

Use evaluations of monomials at sample points to set up incremental linear systems; identify linear dependencies among evaluated monomials to determine leading monomials to remove and to construct a Gröbner basis of the ideal of all polynomials vanishing on the point set.

Demonstration

Given a finite set of points in affine space, form the matrix of monomial evaluations up to a degree bound, perform row reductions to find dependencies, extract polynomials that vanish on the points and continue until a full Gröbner basis and a monomial basis of the coordinate ring are obtained.

Misapplication

Applying Buchberger–Möller without degree or numerical safeguards: using ill-conditioned evaluation matrices or inappropriate monomial ordering can cause unstable linear algebra or failure to detect needed relations, yielding incorrect ideals or missing basis elements.

Consequence

Correct application yields an explicit Gröbner basis of the interpolation ideal and a monomial basis for the coordinate ring, enabling tasks such as polynomial interpolation, solving polynomial systems with known sample sets, and computing multiplicities.

Reversal

The inverse viewpoint is interpolation by symbolic elimination alone (e.g., computing eliminants globally): this may avoid evaluation but typically involves heavier polynomial operations and higher intermediate growth in coefficients.

Boundary

Applies to finite point sets over fields where evaluation is reliable and the vanishing ideal is zero-dimensional; excludes infinite or positive-dimensional varieties and contexts with severe numeric instability unless exact arithmetic is used.

Semantic Tension

There is tension between purely algebraic Gröbner-basis construction (Buchberger's algorithm) and evaluation-based Buchberger–Möller approaches: one trades polynomial arithmetic for linear algebra and must balance numerical stability with algebraic exactness.

Synthesis

The Buchberger–Möller Algorithm builds the vanishing ideal and its Gröbner basis from point evaluations by setting up incremental monomial-evaluation matrices, detecting linear dependencies to produce vanishing polynomials, and iterating until the ideal and coordinate ring basis are determined.