Antonio Macchia

I like to study problems concerning the interactions between Commutative Algebra, Discrete Geometry and Combinatorics.
My PhD dissertation focused on the study of the arithmetical rank (the minimum number of generators up to radical) of squarefree monomial ideals.

1. A combinatorial characterization of S₂ binomial edge ideals (with Davide Bolognini, Giancarlo Rinaldo, Francesco Strazzanti), preprint (2023). [arXiv:2306.17076]
2. Cohen-Macaulay binomial edge ideals of small graphs (with Davide Bolognini, Giancarlo Rinaldo, Francesco Strazzanti), J. Algebra 638 (2024), 189-213, DOI: 10.1016/j.jalgebra.2023.09.029. [arXiv:2212.09181]
   Computations and Algorithm of Section 4 of the paper.
3. General non-realizability certificates for spheres with linear programming (with João Gouveia, Amy Wiebe), J. Symb. Comput. 114 (2023), 172-192, DOI: 10.1016/j.jsc.2022.04.013. [arXiv:2109.15247]
   Non-realizability certificates repository containing the implementation of our algorithm and the non-realizability certificates for the spheres in the paper.
4. Combining realization space models of polytopes (with João Gouveia, Amy Wiebe), Discrete Comput. Geom. 69 (2023), 505-542, DOI: 10.1007/s00454-022-00379-8. [arXiv:2001.11999]
5. Accessible set systems and a conjecture on Cohen-Macaulay binomial edge ideals (with Davide Bolognini, Giancarlo Rinaldo, Francesco Strazzanti), Discrete Mathematics Days 2022, 263 (2022), 49--53, extended abstract, Open Access, DOI: 10.22429/Euc2022.016.
6. Cohen-Macaulay binomial edge ideals and accessible graphs (with Davide Bolognini, Francesco Strazzanti), J. Algebr. Combin. 55 (2022), 1139-1170, Open Access, DOI: 10.1007/s10801-021-01088-w. [arXiv:2101.03619]
7. Powers of monomial ideals with characteristic-dependent Betti numbers (with Davide Bolognini, Francesco Strazzanti, Volkmar Welker), Res. Math. Sci. 9 (2022), article 26, Open Access, DOI: 10.1007/s40687-022-00318-2. [arXiv:2201.00571]
8. A local maximizer for lattice width of 3-dimensional hollow bodies (with Gennadiy Averkov, Giulia Codenotti, Francisco Santos), Discrete Appl. Math. 298 (2021), 7, 129-142, DOI: 10.1016/j.dam.2021.04.009. [arXiv:1907.06199]
9. Regularity of the vanishing ideal over a parallel composition of paths (with Jorge Neves, Maria Vaz Pinto, Rafael H. Villarreal), J. Commut. Algebra 12 (2020), 3, 391-407, DOI: 10.1216/jca.2020.12.391. [arXiv:1606.08621]
10. Slack Ideals in Macaulay2 (with Amy Wiebe), in Mathematical software – ICMS 2020, A. Bigatti, J. Carette, J. Davenport, M. Joswig, T. de Wolff (eds.), 222-231. Lecture Notes in Computer Science, vol. 12097, Springer, Cham, 2020, DOI: 10.1007/978-3-030-52200-1_22. [arXiv:2003.07382]
    SlackIdeals package for Macaulay2 available from version 1.16.
11. Minimal cellular resolutions of the edge ideals of forests (with Margherita Barile), Electron. J. Combin. 27 (2020), 2, paper P.41, 16 pages, DOI: 10.37236/8810. [arXiv:1902.02740]
12. Projectively unique polytopes and toric slack ideals (with João Gouveia, Rekha R. Thomas, Amy Wiebe), J. Pure Appl. Algebra 224 (2020), 5, paper 106229, DOI: 10.1016/j.jpaa.2019.106229. [arXiv:1808.01692]
13. The slack realization space of a polytope (with João Gouveia, Rekha R. Thomas, Amy Wiebe), SIAM J. Discrete Math. 33 (2019), 3, 1637–1653, DOI: 10.1137/18M1233649. [arXiv:1708.04739]
14. On determinantal ideals and algebraic dependence (with Margherita Barile), Comm. Algebra 47 (2019), 6, 2357-2366, DOI: 10.1080/00927872.2018.1492587. [arXiv:1711.01106]
15. A Zen Master, a Zen Monk, a Zen Mathematician (with Sandra Lucente), Nexus Network Journal 20 (2018), 2, 459–474, DOI: 10.1007/s00004-018-0377-9.
16. Binomial edge ideals of bipartite graphs (with Davide Bolognini, Francesco Strazzanti), Eur. J. Combin. 70 (2018), 1-25, DOI: 10.1016/j.ejc.2017.11.004. [arXiv:1704.00152]
17. The poset of proper divisibility (with Davide Bolognini, Emanuele Ventura, Volkmar Welker), SIAM J. Discrete Math. 31 (2017), 3, 2079-2092, DOI: 10.1137/15M1049142. [arXiv:1511.04558]
18. The arithmetical rank of the edge ideals of cactus graphs (with Margherita Barile), Comm. Algebra 45 (2017), 12, 5407–5419, DOI: 10.1080/00927872.2017.1310873. [arXiv:1605.00471]
19. The arithmetical rank of the edge ideals of graphs with pairwise disjoint cycles (with Margherita Barile), J. Algebra Appl. 15 (2016), 7, 22 pages, DOI: 10.1142/S0219498816501206. [arXiv:1511.07310]
20. A note on Cohen-Macaulay graphs (with Margherita Barile), Comm. Algebra 44 (2016), 4, 1473-1477, DOI: 10.1080/00927872.2015.1027361. [pdf]
21. On the ideal of orthogonal representations of a graph in R² (with Jürgen Herzog, Sara Saeedi Madani, Volkmar Welker), Adv. Appl. Math. 71 (2015), 146–173, DOI: 10.1016/j.aam.2015.09.009. [arXiv:1411.3674]
22. Cohomological dimension and arithmetical rank of some determinantal ideals (with Davide Bolognini, Alessio Caminata, Maral Mostafazadehfard), Le Matematiche 70 (2015), 1, 273-300, DOI: 10.4418/2015.70.1.18. [arXiv:1503.06184]
23. Simplicial complexes of whisker type (with Mina Bigdeli, Jürgen Herzog, Takayuki Hibi), Electron. J. Combin. 22 (2015), 1, paper P.75, 10 pages, DOI: 10.37236/4894. [arXiv:1411.7890]
24. The arithmetical rank of the edge ideals of graphs with whiskers, Beitr. Algebra Geom. 56 (2015), 1, 147-158, DOI: 10.1007/s13366-014-0198-2. [arXiv:1307.1652]
25. The set-theoretic complete intersection property for the edge ideals of whisker graphs, Serdica Math. J. 40 (2014), 1, 41–54. [pdf]

• The Importance of Being Alexander Dual, (with Emanuele Ventura), 2012, [pdf]
  Survey on Alexander Duality for the course Combinatorial Commutative Algebra, held by Bernd Sturmfels during Fall 2012, at the University of California, Berkeley
• Gorenstein Rings (Italian), 2012, [pdf]
  Survey on Gorenstein Rings for the course Local Cohomology, held by Margherita Barile during Fall 2011, at Università degli Studi di Bari, Italia

• Ph.D. thesis: The Arithmetical Rank of Edge Ideals, Università degli Studi di Bari, 2014, [pdf]
  Advisor: Prof. Margherita Barile
• Master's thesis: Ideali associati a grafi bipartiti di Cohen-Macaulay, (English title: Edge Ideals of Cohen-Macaulay Bipartite Graphs), Università degli Studi di Bari, 2010
  Advisor: Prof. Margherita Barile
• Bachelor's thesis: Algebre gruppali Lie risolubili e Lie nilpotenti, (English title: Lie Solvable and Lie Nilpotent Group Rings), Università degli Studi di Bari, 2008
  Advisor: Prof. Onofrio Mario Di Vincenzo