Article
| Title: | Some results on ${\rm v}$-number of monomial ideals (English) |
| Author: | Yang, Liuqing |
| Author: | Hu, Kaiwen |
| Author: | Chu, Lizhong |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1311-1331 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We investigate the v-number of various classes of monomial ideals. First, we consider the relationship between the v-number and the regularity of the mixed product ideal $I$, proving that ${\rm v}(I) \leq {\rm reg}(S/I)$. Next, we investigate an open conjecture on the v-number: if a monomial ideal $I$ has linear powers, then for all $k \geq 1$, ${\rm v}(I^k) = \alpha (I)k - 1.$ We prove that if a monomial ideal $I$ with linear powers and $I^k$ (for any $k \geq 1$) has no embedded associated primes, then ${\rm v}(I^k) = \alpha (I)k - 1.$ Additionally, we calculate the \hbox {v-number} of ordinary power and square-free power of edge ideal. Finally, we propose a conjecture that the v-number of ordinary powers of line graph is equal to the v-number of square-free powers. (English) |
| Keyword: | v-number |
| Keyword: | mixed product ideal |
| Keyword: | linear power |
| MSC: | 05E40 |
| MSC: | 13D02 |
| MSC: | 13F55 |
| DOI: | 10.21136/CMJ.2025.0114-25 |
| . | |
| Date available: | 2025-12-20T07:49:35Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153245 |
| . | |
| Reference: | [1] Biswas, P., Mandal, M.: A study of $v$-number for some monomial ideals.Collect. Math. 76 (2025), 667-682 correction ibid. 76 2025 683-684. Zbl 8086181, MR 4950319, 10.1007/s13348-024-00451-x |
| Reference: | [2] Biswas, P., Mandal, M., Saha, K.: Asymptotic behaviour and stability index of $v$-numbers of graded ideals.Available at https://arxiv.org/abs/2402.16583 (2024), 17 pages. 10.48550/arXiv.2402.16583 |
| Reference: | [3] Chu, L., Herzog, J., Lu, D.: The socle module of a monomial ideal.Rocky Mt. J. Math. 51 (2021), 805-821. Zbl 1472.13035, MR 4298829, 10.1216/rmj.2021.51.805 |
| Reference: | [4] Chu, L., Pérez, V. H. Jorge: The Stanley regularity of complete intersections and ideals of mixed products.J. Algebra Appl. 16 (2017), Article ID 1750122, 13 pages. Zbl 1369.13014, MR 3660405, 10.1142/S0219498817501225 |
| Reference: | [5] Civan, Y.: The $v$-number and Castelnuovo-Mumford regularity of graphs.J. Algebraic Comb. 57 (2023), 161-169. Zbl 1508.05180, MR 4544263, 10.1007/s10801-022-01164-9 |
| Reference: | [6] Team, CoCoA: CoCoA: A system for Computations in Commutative Algebra.Available at https://sites.google.com/view/cocoa-cocoalib (2004). |
| Reference: | [7] Cooper, S. M., Seceleanu, A., Tohăneanu, Ş., Pinto, M. Vaz, Villarreal, R. H.: Generalized minimum distance functions and algebraic invariants of Geramita ideals.Adv. Appl. Math. 112 (2020), Article ID 101940, 34 pages. Zbl 1428.13048, MR 4011111, 10.1016/j.aam.2019.101940 |
| Reference: | [8] Crupi, M., Ficarra, A., Lax, E.: Matchings, squarefree powers, and Betti splittings.Ill. J. Math. 69 (2025), 353-372. Zbl 08062121, MR 4919940, 10.1215/00192082-11919226 |
| Reference: | [9] Ficarra, A.: Simon conjecture and the $v$-number of monomial ideals.Available at https://arxiv.org/abs/2309.09188 (2023), 14 pages. 10.48550/arXiv.2309.09188 |
| Reference: | [10] Ficarra, A., Marques, P. C.: The $v$-function of powers of sums of ideals.J. Algebr. Comb. 62 (2025), Article ID 13, 21 pages. Zbl 08080316, MR 4932665, 10.1007/s10801-025-01429-z |
| Reference: | [11] Ficarra, A., Sgroi, E.: Asymptotic behaviour of the $v$-number of homogeneous ideals.Available at https://arxiv.org/abs/2306.14243 (2023), 21 pages. 10.48550/arXiv.2306.14243 |
| Reference: | [12] Fiorindo, L., Ghosh, D.: On the asymptotic behavior of the Vasconcelos invariant for graded modules.Nagoya Math. J. 258 (2025), 296-310. Zbl 8084042, MR 4943018, 10.1017/nmj.2024.33 |
| Reference: | [13] Grisalde, G., Reyes, R. H., Villarreal, R. H.: Induced matchings and the $v$-number of graded ideals.Mathematics 9 (2021), Article ID 2860, 16 pages. 10.3390/math9222860 |
| Reference: | [14] Gu, Y.: Regularity of powers of edge ideals of some graphs.Acta Math. Vietnam. 42 (2017), 445-454. Zbl 1388.13046, MR 3667458, 10.1007/s40306-017-0204-5 |
| Reference: | [15] Jaramillo, D., Villarreal, R. H.: The $v$-number of edge ideals.J. Comb. Theory, Ser. A 177 (2021), Article ID 105310, 35 pages. Zbl 1448.13037, MR 4139109, 10.1016/j.jcta.2020.105310 |
| Reference: | [16] Kumar, M., Nanduri, R., Saha, K.: The slope of the v-function and the Waldschmidt constant.J. Pure Appl. Algebra 229 (2025), Article ID 107881, 13 pages. Zbl 1558.13029, MR 4853484, 10.1016/j.jpaa.2025.107881 |
| Reference: | [17] Martínez-Bernal, J., Pitones, Y., Villarreal, R. H.: Minimum distance functions of graded ideals and Reed-Muller-type codes.J. Pure Appl. Algebra 221 (2017), 251-275. Zbl 1352.13016, MR 3545260, 10.1016/j.jpaa.2016.06.006 |
| Reference: | [18] Núñez-Betancourt, L., Pitones, Y., Villarreal, R. H.: Footprint and minimum distance functions.Commun. Korean Math. Soc. 33 (2018), 85-101. Zbl 1401.13049, MR 3761631, 10.4134/CKMS.c170139 |
| Reference: | [19] Peeva, I.: Graded Syzygies.Algebra and Applications 14. Springer, London (2011). Zbl 1213.13002, MR 2560561, 10.1007/978-0-85729-177-6 |
| Reference: | [20] Rinaldo, G.: Sequentially Cohen-Macaulay mixed product ideals.Algebra Colloq. 22 (2015), 223-232. Zbl 1368.13024, MR 3336057, 10.1142/S1005386715000206 |
| Reference: | [21] Saha, K.: Binomial expansion and the $v$-number.Available at https://arxiv.org/abs/2406.05567 (2024), 7 pages. 10.48550/arXiv.2406.05567 |
| Reference: | [22] Saha, K., Sengupta, I.: The $v$-number of monomial ideals.J. Algebr. Comb. 56 (2022), 903-927. Zbl 1504.13024, MR 4491066, 10.1007/s10801-022-01137-y |
| Reference: | [23] Fakhari, S. A. Seyed: On the Castelnuovo-Mumford regularity of squarefree powers of edge ideals.J. Pure Appl. Algebra 228 (2024), Article ID 107488, 12 pages. Zbl 1537.13029, MR 4624455, 10.1016/j.jpaa.2023.107488 |
| Reference: | [24] Simis, A., Vasconcelos, W. V., Villarreal, R. H.: On the ideal theory of graphs.J. Algebra 167 (1994), 389-416. Zbl 0816.13003, MR 1283294, 10.1006/jabr.1994.1192 |
| Reference: | [25] Vanmathi, A., Sarkar, P.: $v$-numbers of symbolic power filtrations.(to appear) in Collect. Math (2025). 10.1007/s13348-025-00476-w |
| Reference: | [26] Villarreal, R. H.: Monomial Algebras.Pure and Applied Mathematics, Marcel Dekker 238. Marcel Dekker, New York (2001). Zbl 1002.13010, MR 1800904, 10.1201/b18224 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
