### Abstract

Let IF be a field, V ⊆ IF^{n} be a (combinatorially interesting) finite set of points. Several important properties of V are reflected by the polynomial functions on V. To study these, one often considers I(V), the vanishing ideal of V in the polynomial ring IF[x_{1},..., x_{n}]. Gröbner bases and standard monomials of I(V) appear to be useful in this context, leading to structural results on V. Here we survey some work of this type. At the end of the paper a new application of this kind is presented: an algebraic characterization of shattering-extremal families and a fast algorithm to recognize them.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 65-83 |

Number of pages | 19 |

Volume | 6742 LNCS |

DOIs | |

Publication status | Published - 2011 |

Event | 4th International Conference on Algebraic Informatics, CAI 2011 - Linz, Austria Duration: Jun 21 2011 → Jun 24 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6742 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 4th International Conference on Algebraic Informatics, CAI 2011 |
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Country | Austria |

City | Linz |

Period | 6/21/11 → 6/24/11 |

### Fingerprint

### Keywords

- combinatorial Nullstellensatz
- Gröbner basis
- Hilbert function
- inclusion matrix
- lexicographic order
- rank formula
- S-extremal set family
- standard monomial
- vanishing ideal

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 6742 LNCS, pp. 65-83). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6742 LNCS). https://doi.org/10.1007/978-3-642-21493-6_4

**Some combinatorial applications of Gröbner bases.** / Rónyai, L.; Mészáros, Tamás.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 6742 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6742 LNCS, pp. 65-83, 4th International Conference on Algebraic Informatics, CAI 2011, Linz, Austria, 6/21/11. https://doi.org/10.1007/978-3-642-21493-6_4

}

TY - GEN

T1 - Some combinatorial applications of Gröbner bases

AU - Rónyai, L.

AU - Mészáros, Tamás

PY - 2011

Y1 - 2011

N2 - Let IF be a field, V ⊆ IFn be a (combinatorially interesting) finite set of points. Several important properties of V are reflected by the polynomial functions on V. To study these, one often considers I(V), the vanishing ideal of V in the polynomial ring IF[x1,..., xn]. Gröbner bases and standard monomials of I(V) appear to be useful in this context, leading to structural results on V. Here we survey some work of this type. At the end of the paper a new application of this kind is presented: an algebraic characterization of shattering-extremal families and a fast algorithm to recognize them.

AB - Let IF be a field, V ⊆ IFn be a (combinatorially interesting) finite set of points. Several important properties of V are reflected by the polynomial functions on V. To study these, one often considers I(V), the vanishing ideal of V in the polynomial ring IF[x1,..., xn]. Gröbner bases and standard monomials of I(V) appear to be useful in this context, leading to structural results on V. Here we survey some work of this type. At the end of the paper a new application of this kind is presented: an algebraic characterization of shattering-extremal families and a fast algorithm to recognize them.

KW - combinatorial Nullstellensatz

KW - Gröbner basis

KW - Hilbert function

KW - inclusion matrix

KW - lexicographic order

KW - rank formula

KW - S-extremal set family

KW - standard monomial

KW - vanishing ideal

UR - http://www.scopus.com/inward/record.url?scp=79959993101&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959993101&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-21493-6_4

DO - 10.1007/978-3-642-21493-6_4

M3 - Conference contribution

AN - SCOPUS:79959993101

SN - 9783642214929

VL - 6742 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 65

EP - 83

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -