### Abstract

In an earlier work we described Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors v∈ { 0 , 1 } ^{n} of the complete d uniform set family over the ground set [n]. In particular, it turns out that the standard monomials of the above ideal are ballot monomials. We give here a partial extension of this fact. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation a_{1}v_{1}+ ⋯ + a_{n}v_{n}= k, where the a_{i} and k are positive integers. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials, provided that 0 < a_{1}≤ a_{2}≤ ⋯ ≤ a_{n}. As an application, we confirm a conjecture of Frankl in the special case of linear Sperner systems.

Original language | English |
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Article number | 2 |

Journal | Algebra Universalis |

Volume | 79 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 2018 |

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### Keywords

- Ballot monomial
- Characteristic vector
- Gröbner basis
- Polynomial function
- Shattering
- Sperner family
- Standard monomial

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Algebra Universalis*,

*79*(1), [2]. https://doi.org/10.1007/s00012-018-0482-3