Abstract
A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield \({\mathbb {T}}\), the semifield \(\mathbb {Z}_{\mathrm{max}}\) and the two element semifield \({\mathbb {B}}\). The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. It is then shown that the radical of every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension 1. An improvement of a result from Bertram and Easton (Adv Math 308:36–82, 2017) is proven which can be regarded as a Nullstellensatz for tropical polynomials.
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Bertram, A., Easton, R.: The tropical Nullstellensatz for congruences. Adv. Math. 308, 36–82 (2017)
Connes, A., Consani, C.: The arithmetic site. C. R. Math. Ser. I 352, 971–975 (2014)
Connes, A., Consani, C.: Projective geometry in characteristic one and the epicyclic category. Nagoya Math. J. 217, 95–132 (2015)
Giansiracusa, J., Giansiracusa, N.: Equations of tropical varieties, Duke Math. J. 165(18), 3379–3433 (2016)
Izhakian, Z., Rowen, L.: Congruences and coordinate semirings of tropical varieties. Bull. Sci. Math. 140(3), 231–259 (2016)
Lescot, P.: Absolute algebra III—the saturated spectrum. J. Pure Appl. Algebra 216(7), 1004–1015 (2012)
Lorscheid, O.: The geometry of blueprints: part I: algebraic background and scheme theory. Adv. Math. 229(3), 1804–1846 (2012)
Maclagan, D., Rincón, F.: Tropical Schemes, Tropical Cycles, and Valuated Matroids. arXiv:1401.4654
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence, RI (2015)
Mikhalkin, G.: Tropical Geometry and Its Applications, International Congress of Mathematicians, vol. 2, pp. 827–852. Eur. Math. Soc., Zürich (2006). MR 2275625 (2008c:14077)
Robbiano, L.: Term Orderings on the Polynomial Ring, EUROCAL 85, vol. 2 (Linz, 1985), Lecture Notes in Computer Science, vol. 204, pp. 513–517, Springer, Berlin (1985)
Tits, J.: Sur les analogues algébriques des groupes semi-simples complexes, Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre: Centre Belge de Recherches Mathématiques Établissements Ceuterick. Louvain; Librairie Gauthier-Villars, Paris, pp. 261–289 (1957)
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Dániel Joó: This research was partially supported by National Research, Development and Innovation Office, NKFIH K 119934, NKFIH PD 121410 and the exchange project “Combinatorial ring theory” between the Bulgarian and Hungarian Academies of Sciences.
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Joó, D., Mincheva, K. Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials. Sel. Math. New Ser. 24, 2207–2233 (2018). https://doi.org/10.1007/s00029-017-0322-x
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DOI: https://doi.org/10.1007/s00029-017-0322-x