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Gröbner Bases of Toric Ideals Associated with Matroids

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Abstract

In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gröbner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gröbner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange.

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Acknowledgements

The authors would like to thank professor Hidefumi Ohsugi for useful comments and suggestions.

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan

    Ken-ichi Hayase, Takayuki Hibi & Koyo Katsuno

  2. Department of Mathematics, Collage of Science, Rikkyo University, Toshima-ku, Tokyo, 171-8501, Japan

    Kazuki Shibata

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  1. Ken-ichi Hayase
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  2. Takayuki Hibi
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  3. Koyo Katsuno
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  4. Kazuki Shibata
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Correspondence to Kazuki Shibata.

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Hayase, Ki., Hibi, T., Katsuno, K. et al. Gröbner Bases of Toric Ideals Associated with Matroids. Acta Math Vietnam 47, 775–779 (2022). https://doi.org/10.1007/s40306-021-00468-5

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  • DOI: https://doi.org/10.1007/s40306-021-00468-5

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