×
Kala, Vítězslav

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

Commentat. Math. Univ. Carol. 58, No. 3, 267-273 (2017).
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

Keywords:

Abhyankar’s construction; semiring; semifield; finitely generated; additively idempotent

Citations:

Zbl 1227.14004
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Abhyankar S. S., Pillars and towers of quadratic transformations, Proc. Amer. Math. Soc. 139 (2011), 3067-3082 · Zbl 1227.14004 · doi:10.1090/S0002-9939-2011-10731-7
[2] El Bashir R.; Hurt J.; Jančařík A.; Kepka T., Simple commutative semirings, J. Algebra 236 (2001), 277-306 · Zbl 0976.16034 · doi:10.1006/jabr.2000.8483
[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 · doi:10.1515/form.2011.059
[4] Di Nola A.; Gerla B., Algebras of Lukasiewicz’s logic and their semiring reducts, Contemp. Math. 377 (2005), 131-144 · Zbl 1081.06009 · doi:10.1090/conm/377/06988
[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 · doi:10.1007/BF01057937
[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 · doi:10.1515/form.2011.068
[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 · doi:10.1515/ms-2016-0203
[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 · doi:10.1142/S0219498817500384
[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 · doi:10.1007/s00233-016-9827-4
[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 · doi:10.1007/s00208-017-1527-1
[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 · doi:10.1016/0022-1236(86)90015-7
[19] Weinert H. J., Über Halbringe und Halbkörper I, Acta Math. Acad. Sci. Hungar. 13 (1962), 365-378 · Zbl 0125.01002 · doi:10.1007/BF02020799
[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 · doi:10.1007/BF01879738
[21] Zumbrägel J., Public-key cryptography based on simple semirings, PhD Thesis, Universität Zürich, Switzerland, 2008
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.