An algebraist’s view on border bases

Summary

This chapter is devoted to laying the algebraic foundations for border bases of ideals. Using an order ideal $\mathcal{O}$, we describe a zero-dimensional ideal from the outside. The first and higher borders of $\mathcal{O}$ can be used to measure the distance of a term from $\mathcal{O}$ and to define $\mathcal{O}$-border bases. We study their existence and uniqueness, their relation to Gröbner bases, and their characterization in terms of commuting matrices. Finally, we use border bases to solve a problem coming from statistics.