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Semifields and a theorem of Abhyankar. (English) Zbl 1434.12009

Summary: S. S. Abhyankar [Proc. Am. Math. Soc. 139, No. 9, 3067–3082 (2011; Zbl 1227.14004)] proved that every field of finite transcendence degree over \(\mathbb{Q}\) or over a finite field is a homomorphic image of a subring of the ring of polynomials \(\mathbb{Z}[T_1,\dots ,T_n]\) (for some \(n\) depending on the field). We conjecture that his result cannot be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings.

MSC:

12K10 Semifields
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13B25 Polynomials over commutative rings
16Y60 Semirings

Citations:

Zbl 1227.14004
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References:

[1] Abhyankar S. S., Pillars and towers of quadratic transformations, Proc. Amer. Math. Soc. 139 (2011), 3067-3082 · Zbl 1227.14004
[2] El Bashir R.; Hurt J.; Jančařík A.; Kepka T., Simple commutative semirings, J. Algebra 236 (2001), 277-306 · Zbl 0976.16034
[3] Busaniche M.; Cabrer L.; Mundici D., Confluence and combinatorics in finitely generated unital lattice-ordered abelian groups, Forum Math. 24 (2012), 253-271 · Zbl 1277.06007
[4] Di Nola A.; Gerla B., Algebras of Lukasiewicz’s logic and their semiring reducts, Contemp. Math. 377 (2005), 131-144 · Zbl 1081.06009
[5] Di Nola A.; Lettieri A., Perfect MV-algebras are categorically equivalent to abelian \(\ell \)-groups, Studia Logica 53 (1994), 417-432 · Zbl 0812.06010
[6] Golan J. S., Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999 · Zbl 0947.16034
[7] Ježek J.; Kala V.; Kepka T., Finitely generated algebraic structures with various divisibility conditions, Forum Math. 24 (2012), 379-397 · Zbl 1254.16041
[8] Kala V., Lattice-ordered groups finitely generated as semirings, J. Commut. Alg., to appear, 16 pp., arxiv:1502.01651 · Zbl 1373.06022
[9] Kala V.; Kepka T., A note on finitely generated ideal-simple commutative semirings, Comment. Math. Univ. Carolin. 49 (2008), 1-9 · Zbl 1192.16045
[10] Kala V.; Kepka T.; Korbelář M., Notes on commutative parasemifields, Comment. Math. Univ. Carolin. 50 (2009), 521-533 · Zbl 1203.16038
[11] Kala V.; Korbelář M., Idempotence of commutative semifields, preprint, 16 pp · Zbl 1404.12010
[12] Kepka T.; Korbelář M., Conjectures on additively divisible commutative semirings, Math. Slovaca 66 (2016), 1059-1064 · Zbl 1399.16118
[13] Korbelář M.; Landsmann G., One-generated semirings and additive divisibility, J. Algebra Appl. 16 (2017), 1750038, 22 pp., DOI: 10.1142/S0219498817500384 · Zbl 1358.16038
[14] Korbelář M., Torsion and divisibility in finitely generated commutative semirings, Semigroup Forum, to appear; DOI: 10.1007/s00233-016-9827-4 · Zbl 1453.16052
[15] Leichtnam E., A classification of the commutative Banach perfect semi-fields of characteristic 1. Applications, to appear in Math. Ann., DOI: 10.1007/s00208-017-1527-1 · Zbl 1374.46053
[16] Litvinov G. L., The Maslov dequantization, idempotent and tropical mathematics: a brief introduction, arXiv:math/0507014 · Zbl 1102.46049
[17] Monico C. J., Semirings and semigroup actions in public-key cryptography, PhD Thesis, University of Notre Dame, USA, 2002
[18] Mundici D., Interpretation of AF \(C^*\)-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15-63 · Zbl 0597.46059
[19] Weinert H. J., Über Halbringe und Halbkörper I, Acta Math. Acad. Sci. Hungar. 13 (1962), 365-378 · Zbl 0125.01002
[20] Weinert H. J.; Wiegandt R., On the structure of semifields and lattice-ordered groups, Period. Math. Hungar. 32 (1996), 147-162 · Zbl 0896.12001
[21] Zumbrägel J., Public-key cryptography based on simple semirings, PhD Thesis, Universität Zürich, Switzerland, 2008
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