Gröbner bases are used in experimental design and interpolation. They provide a special way to write a finite system of polynomial equations so that favourable interpretation of properties such as interpolation is available. The polynomial interpolators give a neat generalisation of Newton's divided difference interpolation formula. In this paper, these basic ideas are applied to general grids using computational commutative algebra and algebraic geometry. A multivariate generalised divided difference formula is given for an arbitrary set of points with no subsets of three points lying on a line. The obtained result is an extension of the Newton polynomials and the Newton interpolation formula. The generalisation is based on Gröbner bases for the grid expressed as a zero-dimensional variety and is dependent on the chosen term ordering and the selected ordering points in the grid.
Polynomial ideals, monomial bases, and a divided difference formula
RICCOMAGNO, EVA;
2005-01-01
Abstract
Gröbner bases are used in experimental design and interpolation. They provide a special way to write a finite system of polynomial equations so that favourable interpretation of properties such as interpolation is available. The polynomial interpolators give a neat generalisation of Newton's divided difference interpolation formula. In this paper, these basic ideas are applied to general grids using computational commutative algebra and algebraic geometry. A multivariate generalised divided difference formula is given for an arbitrary set of points with no subsets of three points lying on a line. The obtained result is an extension of the Newton polynomials and the Newton interpolation formula. The generalisation is based on Gröbner bases for the grid expressed as a zero-dimensional variety and is dependent on the chosen term ordering and the selected ordering points in the grid.
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