Course Materials

Course (Topic 1):  Border Bases

Abstract: An ideal I in the polynomial ring P=K[x_1,...,x_n] over a field K is called 0-dimensional if the residue class ring P/I is a finite dimensional K-vector space. Equivalently, the polynomials in the ideal I have only finitely many common zeros over the algebraic closure of K. For such ideals, border bases are a very versatile and far-reaching generalization of Gröbner bases. A border basis of I is a special set of generators of I associated to an order ideal O, i.e., to a divisor-closed set of terms. In this course we use CoCoA to study the basic theory of border bases, their computation, and two of their main applications.

The first application is the possibility to consider the family of all 0-dimensional ideals having an O-border basis. This family is parametrized by the border basis scheme B_O. Border basis schemes are examples of moduli spaces in algebraic geometry. They form open coverings of Hilbert schemes and have the advantage of an explicit and easily computable desciption. We also look at special subschemes of B_O, for instance corresponding to homogeneous ideals, to locally Gorenstein schemes, or to strict Gorenstein schemes.

The second application is to solve 0-dimensional polynomial systems. In addition to the exact case, border bases also offer a greatly improved numerical stability compared to Gröbner bases, as well as the possibility to calculate approximate models of physical systems.

The content of the course is outlined as follows:

  1. Introduction to Border Bases
    • Zero-dimensional affine algebras
    • The Finiteness Criterion
    • The Buchberger-Möller Algorithm
    • Definition of border bases
    • Characterizing border bases using commuting matrices
  2. Characterizations of Border Bases
    • The Border Division Algorithm
    • Characterizing border bases using border form Ideals and rewrite rules
    • The Buchberger criterion for border bases
    • Neighbour syzygies
  3.  The Border Basis Scheme
    • The definition of border basis schemes
    • The total arrow grading
    • Homogeneous border basis schemes
  4.  Subschemes of the Border Basis Scheme
    • The canonical module of a 0-dimensional scheme
    • Locally and strict Gorenstein rings
    • The locally Gorenstein locus
    • The degree filtered border basis scheme
    • The strict Gorenstein locus
  5.  Computing Border Bases
    • The Border Basis Algorithm
    • Approximate data
    • Approximate border bases
    • Singular value decompositions
    • The AVI Algorithm


[1] M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer-Verlag, Berlin, 2000.

[2] M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer-Verlag, Berlin, 2005.  

[3] M. Kreuzer and L. Robbiano, Computational Linear and Commutative Algebra, Springer Int. Publ., Cham, 2016.

[4] J. Abbott and L. Robbiano (eds.), Approximate Commutative Algebra, Springer-Verlag, Vienna, 2009.

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Course (Topic 2):  Binomial Ideals

(1) An introduction to binomial and toric ideals

Abstract:  In my  lectures, binomial ideals and  in particular toric ideals associated with a configuration matrix are considered and their basic properties are shown. Binomial prime ideals will be characterized and it will be shown that the reduced Gröbner basis of a binomial ideal is again a binomial ideal. We consider the Graver basis of a monomial ideal and relate it to its universal Gröbner basis. We also consider lattice ideals,  lattice basis ideals and Lawrence ideals which allow us to describe a method to compute the Graver basis of a lattice ideal. Finally we give a survey on  special classes of binomial ideals which include binomial edge ideals and the ideal of inner minore of a polyomino.

  • Toric Ideals and Binomial Ideals
  • Lattice and Lattice Basis Ideals
  • Lawrence Ideals
  • Binomial Edge Ideals and Polyominoes
  • The Squarefree Divisor Complex

(2) Five lectures on Hibi rings

Abstract:  The Hibi ring of a finite distributive lattice was introduced in [2].  Over the past two decades, many papers and many authors have deeply studied Hibi rings from various viewpoints of, say, finite partially ordered sets, convex polytopes, commutative algebra together with representation theory.  The series of my talks will discuss the fundamental materials on Hibi rings.  No special knowledge will be required to understand my talks. The content is organized as follows:

  • Partially ordered sets and distributive lattices
    • Finite partially ordered sets and finite lattices
    • Birkhoff's fundamental structure theorem for finite distributive lattices
    • Dedekind's theorem
  • Hibi rings and their h-polynomials
    • Definition of Hibi rings
    • Order preserving maps and strictly order preserving maps
    • The h-polynomial of a Hibi ring
  •  Binomial ideals of finite lattices
    • The defining ideal of a Hibi ring
    • The reduced Gröbner basis of a Hibi ring
    • Characterization of planar distributive lattices
  • Order polytopes and their toric rings
    • Order polytopes and chain polytopes
    • Unimodular equivalence
    • The toric ring of an order polytope
  • Hibi ideals and their Alexander dual
    • Definition of Hibi ideals
    • The Alexander dual of a Hibi ideal
    • Cohen--Macaulay bipartite graphs


[1] J. Herzog and T. Hibi, Monomial Ideals, Graduate Texts in Mathematics 260, Springer, 2011.  

[2] T. Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws, in: Commutative Algebra and Combinatorics (M. Nagata and H. Matsumura, Eds.), Advanced Studies in Pure Math., Volume 11, North--Holland, Amsterdam, 1987, pp. 93-109.

[3] T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, Glebe, N.S.W., Australia, 1992. 

[4] J. Herzog,  T. Hibi  and H. Ohsugi, Binomial ideals, Graduate Texts in Mathematics 279, Springer, Cham, 2018.

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Poster Session

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