Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
v-number; mixed product ideal; linear power
v-number; mixed product ideal; linear power
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.
References:
[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. DOI 10.1007/s13348-024-00451-x | MR 4950319 | Zbl 8086181
[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. DOI 10.48550/arXiv.2402.16583
[3] Chu, L., Herzog, J., Lu, D.: The socle module of a monomial ideal. Rocky Mt. J. Math. 51 (2021), 805-821. DOI 10.1216/rmj.2021.51.805 | MR 4298829 | Zbl 1472.13035
[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. DOI 10.1142/S0219498817501225 | MR 3660405 | Zbl 1369.13014
[5] Civan, Y.: The $v$-number and Castelnuovo-Mumford regularity of graphs. J. Algebraic Comb. 57 (2023), 161-169. DOI 10.1007/s10801-022-01164-9 | MR 4544263 | Zbl 1508.05180
[6] Team, CoCoA: CoCoA: A system for Computations in Commutative Algebra. Available at https://sites.google.com/view/cocoa-cocoalib (2004).
[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. DOI 10.1016/j.aam.2019.101940 | MR 4011111 | Zbl 1428.13048
[8] Crupi, M., Ficarra, A., Lax, E.: Matchings, squarefree powers, and Betti splittings. Ill. J. Math. 69 (2025), 353-372. DOI 10.1215/00192082-11919226 | MR 4919940 | Zbl 08062121
[9] Ficarra, A.: Simon conjecture and the $v$-number of monomial ideals. Available at https://arxiv.org/abs/2309.09188 (2023), 14 pages. DOI 10.48550/arXiv.2309.09188
[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. DOI 10.1007/s10801-025-01429-z | MR 4932665 | Zbl 08080316
[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. DOI 10.48550/arXiv.2306.14243
[12] Fiorindo, L., Ghosh, D.: On the asymptotic behavior of the Vasconcelos invariant for graded modules. Nagoya Math. J. 258 (2025), 296-310. DOI 10.1017/nmj.2024.33 | MR 4943018 | Zbl 8084042
[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. DOI 10.3390/math9222860
[14] Gu, Y.: Regularity of powers of edge ideals of some graphs. Acta Math. Vietnam. 42 (2017), 445-454. DOI 10.1007/s40306-017-0204-5 | MR 3667458 | Zbl 1388.13046
[15] Jaramillo, D., Villarreal, R. H.: The $v$-number of edge ideals. J. Comb. Theory, Ser. A 177 (2021), Article ID 105310, 35 pages. DOI 10.1016/j.jcta.2020.105310 | MR 4139109 | Zbl 1448.13037
[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. DOI 10.1016/j.jpaa.2025.107881 | MR 4853484 | Zbl 1558.13029
[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. DOI 10.1016/j.jpaa.2016.06.006 | MR 3545260 | Zbl 1352.13016
[18] Núñez-Betancourt, L., Pitones, Y., Villarreal, R. H.: Footprint and minimum distance functions. Commun. Korean Math. Soc. 33 (2018), 85-101. DOI 10.4134/CKMS.c170139 | MR 3761631 | Zbl 1401.13049
[19] Peeva, I.: Graded Syzygies. Algebra and Applications 14. Springer, London (2011). DOI 10.1007/978-0-85729-177-6 | MR 2560561 | Zbl 1213.13002
[20] Rinaldo, G.: Sequentially Cohen-Macaulay mixed product ideals. Algebra Colloq. 22 (2015), 223-232. DOI 10.1142/S1005386715000206 | MR 3336057 | Zbl 1368.13024
[21] Saha, K.: Binomial expansion and the $v$-number. Available at https://arxiv.org/abs/2406.05567 (2024), 7 pages. DOI 10.48550/arXiv.2406.05567
[22] Saha, K., Sengupta, I.: The $v$-number of monomial ideals. J. Algebr. Comb. 56 (2022), 903-927. DOI 10.1007/s10801-022-01137-y | MR 4491066 | Zbl 1504.13024
[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. DOI 10.1016/j.jpaa.2023.107488 | MR 4624455 | Zbl 1537.13029
[24] Simis, A., Vasconcelos, W. V., Villarreal, R. H.: On the ideal theory of graphs. J. Algebra 167 (1994), 389-416. DOI 10.1006/jabr.1994.1192 | MR 1283294 | Zbl 0816.13003
[25] Vanmathi, A., Sarkar, P.: $v$-numbers of symbolic power filtrations. (to appear) in Collect. Math (2025). DOI 10.1007/s13348-025-00476-w
[26] Villarreal, R. H.: Monomial Algebras. Pure and Applied Mathematics, Marcel Dekker 238. Marcel Dekker, New York (2001). DOI 10.1201/b18224 | MR 1800904 | Zbl 1002.13010
Search
Partner of
