Basis condition for generalized spline modules. (English) Zbl 1533.05234
Summary: A generalized spline on an edge-labeled graph \((G,\alpha)\) is defined as a vertex labeling, such that the difference of labels on adjacent vertices lies in the ideal generated by the edge label. We study generalized splines over greatest common divisor domains and present a determinantal basis condition for generalized spline modules on arbitrary graphs. The main result of the paper answers a conjecture that appeared in several papers.
References:
| [1] | Selma, A., Katie, A., Daniel, A., Luisa, A., Chloe, I., Samet, S., and Luke, S.: Generalized graph splines and the universal difference property. arXiv e-prints, arXiv: 2206.06981, (2022) |
| [2] | Altınok, S.; Dilaver, G., Minimum generating sets for complete graphs, TWMS J. Appl. Eng. Math., 13, 3, 1239-1253, 2023 |
| [3] | Selma, Altınok, and Samet, Sarıoǧlan.: Basis criteria for generalized spline modules via determinant. Discrete Mathematics. 344(2) (2021) · Zbl 1453.05111 |
| [4] | Selma, Altınok, and Samet, Sarıoǧlan.: Flow-up bases for generalized spline modules on arbitrary graphs. Journal of Algebra and Its Applications, 20(10) (2021) · Zbl 1476.05176 |
| [5] | Anders, K.; Crans, A.; Foster-Greenwood, B.; Mellor, B.; Tymoczko, J., Graphs admitting only constant splines, Pac. J. Math., 304, 2, 385-400, 2020 · Zbl 1481.13014 · doi:10.2140/pjm.2020.304.385 |
| [6] | Portia, Anderson, Jacob, P. Matherne, and Julianna, Tymoczko.: Generalized splines on graphs with two labels and polynomial splines on cycles. arXiv e-prints, arXiv: 2108.02757 (2021) · Zbl 1537.41005 |
| [7] | Billera, LJ; Rose, LL, Modules of piecewise polynomials and their freeness, Math. Z., 209, 485-497, 1992 · Zbl 0891.13004 · doi:10.1007/BF02570848 |
| [8] | Kathryn Elizabeth Blaine.: Determinantal conditions on integer splines. Bard College Senior Projects Fall 2018 (2018) |
| [9] | Bowden, N., Hagen, S., King, M., Reinders, S.: Bases and structure constants of generalized splines with integer coefficients on cycles. arXiv e-prints, arXiv:1502.00176 (2015) |
| [10] | Nealy, Bowden, and Julianna, Tymoczko.: Splines mod \(m\). arXiv e-prints, arXiv:1501.02027 (2015) |
| [11] | Kariane, Calta, and Lauren, L. Rose.: Determinantal conditions for modules of generalized splines. Journal of Algebra and Its Applications, page 2450236 |
| [12] | DiPasquale, M., Generalized splines and graphic arrangements, J. Algebraic Combin., 45, 171-189, 2017 · Zbl 1364.13028 · doi:10.1007/s10801-016-0704-8 |
| [13] | Radha, Madhavi Duggaraju, and Lipika, Mazumdar.: An algorithm for generating generalized splines on graphs such as complete graphs, complete bipartite graphs and hypercubes. AKCE International Journal of Graphs and Combinatorics (2019) · Zbl 1473.05265 |
| [14] | Gilbert, S.; Tymoczko, J.; Viel, S., Generalized splines on arbitrary graphs, Pac. J. Math., 281, 333-364, 2016 · Zbl 1331.05183 · doi:10.2140/pjm.2016.281.333 |
| [15] | Ester, Gjoni.: Basis criteria for \(n\)-cycle integer splines. Bard College Senior Projects Spring 2015 (2015) |
| [16] | Mark, Goresky, Robert, E. Kottwitz, and Macpherson, R. D.: Equivariant cohomology, koszul duality, and the localization theorem. Inventiones mathematicae, 131, 25-83 (1997) · Zbl 0897.22009 |
| [17] | Madeline, Handschy, Julie, Melnick, and Stephanie, Reinders.: Integer generalized splines on cycles. arXiv e-prints, arXiv:1409.1481 (2014) |
| [18] | Emmet Reza, Mahdavi.: Integer generalized splines on the diamond graph. Bard College Senior Projects Spring 2016 (2016) |
| [19] | Orlik, P.; Terao, H., Arrangements of Hyperplanes, 1992, Berlin Heidelberg: Grundlehren der mathematischen Wissenschaften. Springer, Berlin Heidelberg · Zbl 0757.55001 · doi:10.1007/978-3-662-02772-1 |
| [20] | McCleary, Philbin, Lindsay, Swift, Alison, Tammaro, and Danielle, Williams.: Splines over integer quotient rings. arXiv e-prints, arXiv:1706.00105 (2017) |
| [21] | Rose, LL, Module bases for multivariate splines, J. Approx. Theory, 86, 1, 13-20, 1996 · Zbl 0869.41009 · doi:10.1006/jath.1996.0052 |
| [22] | Rose, LL; Suzuki, J., Generalized integer splines on arbitrary graphs, Discret. Math., 346, 1, 2023 · Zbl 1521.05176 · doi:10.1016/j.disc.2022.113139 |
| [23] | Kyoji, Saito.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math, 27(2), 265-291 (1980) · Zbl 0496.32007 |
| [24] | Samet, Sarıoǧlan, and Selma, Altınok.: Multivariate generalized splines and syzygies on graphs. Journal of Algebra and Its Applications, page 2450039 · Zbl 1539.13029 |
| [25] | Schenck, H., Equivariant chow cohomology of nonsimplicial toric varieties, Trans. Am. Math. Soc., 364, 8, 4041-4051, 2012 · Zbl 1408.14166 · doi:10.1090/S0002-9947-2012-05409-2 |
| [26] | Tymoczko, J., An introduction to equivariant cohomology and homology, Contemp. Math., 388, 169-188, 2005 · doi:10.1090/conm/388/07264 |
| [27] | Tymoczko, J., Splines in geometry and topology, Computer Aided Geometric Design, 45, 32-47, 2016 · Zbl 1418.41013 · doi:10.1016/j.cagd.2015.11.006 |
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