Kehrein, Achim; Kreuzer, Martin; Robbiano, Lorenzo An algebraist’s view on border bases. (English) Zbl 1152.13304 Dickenstein, Alicia (ed.) et al., Solving polynomial equations. Foundations, algorithms, and applications. Berlin: Springer (ISBN 3-540-24326-7/hbk). Algorithms and Computation in Mathematics 14, 169-202, 393-418 (2005). 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.For the entire collection see [Zbl 1061.12001]. Cited in 1 ReviewCited in 24 Documents MSC: 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) PDFBibTeX XMLCite \textit{A. Kehrein} et al., Algorithms Comput. Math. 14, 169--202, 393--418 (2005; Zbl 1152.13304)