From algebraic sets to monomial linear bases by means of combinatorial algorithms (Q1893979)

Let \(K\) be a field and let \(A = K[x_1, \ldots, x_n]/J\) be a finite dimensional \(K\)-algebra. A minimal monomial linear base of \(A\) which is minimal (in a suitable sense and with respect to a chosen term-order) induces, in a natural way, an \(n\)-dimensional Ferrers diagram (i.e., a finite ideal in the standard poset structure of \(\mathbb{N}^n)\). If \({\mathcal P} \subset K^n\) is a finite set of points, then the algebra \(K[x_1, \ldots, x_n]/J\) is finite dimensional over \(K\), where \(J : = {\mathcal I} ({\mathcal P}) \subset K [x_1, \ldots, x_n]\) is the ideal of vanishing polynomials on \({\mathcal P}\). The authors give a combinatorial algorithm to obtain from \({\mathcal P}\) an \(n\)-dimensional Ferrers diagram whose corresponding set of monomials is a minimal monomial linear basis with respect to the inverse lexicographic order induced by \(x_1 < \cdots < x_n\). They also state a more or less straightforward generalization to the case of an algebraic multiset in place of \({\mathcal P}\). There are some applications which include obtaining a reduced Gröbner basis of the ideal \(J\) with respect to the inverse lexicographic order and finding an interpolation polynomial in a suitable generalization of the traditional Lagrange-Hermite interpolation problem.