Multivariate Interpolation and Standard Bases for Macaulay Modules

The -linear space of all n-linearly recursive functions (1.1) (=evaluated differential forms) for which a zero-dimensional ideal [x1, …, xn] is the largest ideal which is contained in the kernel of all of them turns out to be the orthogonal -space ( [x1, …, xn])* of and is known as Macaulayʹs inverse system of . Making use of the antiderivative operator , the whole space of all differential forms can be endowed with a structure of [x1, …, xn]-module; with respect to finitely generated submodules of it (which we call Macaulay modules), we describe a dual analog of the Gröbner bases theory. The motivation for studying Macaulay modules has to be found mainly in multivariate interpolation problems and in the theory of polynomial bialgebras, though some application to algebraic geometry is not excluded.