×

zbMATH — the first resource for mathematics

Tropical ideals. (English) Zbl 1428.14093
Summary: We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing J. Giansiracusa and N. Giansiracusa [Duke Math. J. 165, No. 18, 3379–3433 (2016; Zbl 1409.14100)]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

MSC:
14T10 Foundations of tropical geometry and relations with algebra
05B35 Combinatorial aspects of matroids and geometric lattices
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ardila, F. and Billey, S., Flag arrangements and triangulations of products of simplices, Adv. Math.214 (2007), 495-524. doi:10.1016/j.aim.2007.02.014 · Zbl 1194.14078
[2] Baker, M., Matroids over hyperfields, Preprint (2016), arXiv:1601.01204v3.
[3] Bertram, A. and Easton, R., The tropical Nullstellensatz for congruences, Adv. Math.308 (2017), 36-82. doi:10.1016/j.aim.2016.12.004 · Zbl 1407.14058
[4] Bruns, W. and Herzog, J., Cohen-Macaulay rings, (Cambridge University Press, Cambridge, 1993).
[5] Cools, F., Draisma, J., Payne, S. and Robeva, E., A tropical proof of the Brill-Noether theorem, Adv. Math.230 (2012), 759-776. doi:10.1016/j.aim.2012.02.019 · Zbl 1325.14080
[6] Cox, D. A., Little, J. B. and Schenck, H. K., Toric varieties, (American Mathematical Society, Providence, RI, 2011). doi:10.1090/gsm/124 · Zbl 1223.14001
[7] Dress, A. and Wenzel, W., Valuated matroids, Adv. Math.93 (1992), 214-250. doi:10.1016/0001-8708(92)90028-J · Zbl 0754.05027
[8] Durov, N., New approach to Arakelov geometry, Preprint (2007), arXiv:0704.2030. · Zbl 1141.14301
[9] Giansiracusa, J. and Giansiracusa, N., Equations of tropical varieties, Duke Math. J.165 (2016), 3379-3433. doi:10.1215/00127094-3645544 · Zbl 1409.14100
[10] Grigg, N. and Manwaring, N., An elementary proof of the fundamental theory of tropical algebra, Preprint (2007), arXiv:0707.2591.
[11] Grigoriev, D. and Podolskii, V. V., Tropical effective primary and dual Nullstellensätze, Preprint (2014), arXiv:1409.6215.
[12] Gross, M., Tropical geometry and mirror symmetry, (American Mathematical Society, Providence, RI, 2011), published for the Conference Board of the Mathematical Sciences, Washington, DC. doi:10.1090/cbms/114 · Zbl 1215.14061
[13] Haque, M. M., Tropical incidence relations, polytopes, and concordant matroids, Preprint (2012), arXiv:1211.2841.
[14] Itenberg, I., Katzarkov, L., Mikhalkin, G. and Zharkov, I., Tropical homology, Preprint (2016),arXiv:1604.01838.
[15] Jensen, D. and Payne, S., Tropical independence I: shapes of divisors and a proof of the Gieseker-Petri theorem, Algebra Number Theory8 (2014), 2043-2066. doi:10.2140/ant.2014.8.2043 · Zbl 1317.14139
[16] Jensen, D. and Payne, S., Tropical independence II: the maximal rank conjecture for quadrics, Algebra Number Theory10 (2016), 1601-1640. doi:10.2140/ant.2016.10.1601 · Zbl 1379.14020
[17] Joó, D. and Mincheva, K., Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials, Selecta Math. (NS) (2017), doi:10.1007/s00029-017-0322-x.
[18] Kajiwara, T., Tropical toric geometry, in Toric topology, (American Mathematical Society, Providence, RI, 2008), 197-207. doi:10.1090/conm/460/09018 · Zbl 1202.14047
[19] Lorscheid, O., The geometry of blueprints: Part I: algebraic background and scheme theory, Adv. Math., 229, 1804-1846, (2012) · Zbl 1259.14001
[20] Lorscheid, O., Scheme theoretic tropicalization, Preprint (2015), arXiv:1508.07949.
[21] Lorscheid, O., Blue schemes, semiring schemes, and relative schemes after Toën and Vaquié, J. Algebra, 482, 264-302, (2017) · Zbl 1401.14015
[22] Maclagan, D., Antichains of monomial ideals are finite, Proc. Amer. Math. Soc.129 (2001), 1609-1615; (electronic). doi:10.1090/S0002-9939-00-05816-0 · Zbl 0984.13013
[23] Maclagan, D. and Rincón, F., Tropical schemes, tropical cycles, and valuated matroids, J. Eur. Math. Soc. (JEMS), to appear. Preprint (2014), arXiv:1401.4654.
[24] Maclagan, D. and Sturmfels, B., Introduction to Tropical Geometry, (American Mathematical Society, Providence, RI, 2015). doi:10.1090/gsm/161 · Zbl 1321.14048
[25] Mikhalkin, G., Enumerative tropical algebraic geometry in ℝ2, J. Amer. Math. Soc., 18, 313-377, (2005) · Zbl 1092.14068
[26] Miller, E. and Sturmfels, B., Combinatorial commutative algebra, (Springer, New York, 2005). · Zbl 1090.13001
[27] Murota, K., Matrices and matroids for systems analysis, (2010), Springer: Springer, Berlin · Zbl 1181.05001
[28] Murota, K. and Tamura, A., On circuit valuation of matroids, Adv. Appl. Math.26 (2001), 192-225. doi:10.1006/aama.2000.0716 · Zbl 0979.05028
[29] Oxley, J. G., Matroid theory, (1992), Oxford Science Publications, Clarendon Press, Oxford University Press: Oxford Science Publications, Clarendon Press, Oxford University Press, New York · Zbl 0784.05002
[30] Payne, S., Analytification is the limit of all tropicalizations, Math. Res. Lett., 16, 543-556, (2009) · Zbl 1193.14077
[31] Rabinoff, J., Tropical analytic geometry, Newton polygons, and tropical intersections, Adv. Math., 229, 3192-3255, (2012) · Zbl 1285.14072
[32] Shustin, E. and Izhakian, Z., A tropical Nullstellensatz, Proc. Amer. Math. Soc. (2007), 3815-3821. doi:10.1090/S0002-9939-07-09005-3 · Zbl 1163.12004
[33] Toën, B. and Vaquié, M., Au-dessous de Spec ℤ, J. K-Theory3 (2009), 437-500. doi:10.1017/is008004027jkt048 · Zbl 1177.14022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.