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.