| Title:
|
Semifields and a theorem of Abhyankar (English) |
| Author:
|
Kala, Vítězslav |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
58 |
| Issue:
|
3 |
| Year:
|
2017 |
| Pages:
|
267-273 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Abhyankar 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. (English) |
| Keyword:
|
Abhyankar's construction |
| Keyword:
|
semiring |
| Keyword:
|
semifield |
| Keyword:
|
finitely generated |
| Keyword:
|
additively idempotent |
| MSC:
|
12K10 |
| MSC:
|
13B25 |
| MSC:
|
16Y60 |
| idZBL:
|
Zbl 06837064 |
| idMR:
|
MR3708772 |
| DOI:
|
10.14712/1213-7243.2015.216 |
| . |
| Date available:
|
2017-11-22T09:17:23Z |
| Last updated:
|
2019-10-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146905 |
| . |
| Reference:
|
[1] Abhyankar S.S.: Pillars and towers of quadratic transformations.Proc. Amer. Math. Soc. 139 (2011), 3067–3082. Zbl 1227.14004, MR 2811263, 10.1090/S0002-9939-2011-10731-7 |
| Reference:
|
[2] El Bashir R., Hurt J., Jančařík A., Kepka T.: Simple commutative semirings.J. Algebra 236 (2001), 277–306. Zbl 0976.16034, MR 1808355, 10.1006/jabr.2000.8483 |
| Reference:
|
[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, MR 2900005, 10.1515/form.2011.059 |
| Reference:
|
[4] Di Nola A., Gerla B.: Algebras of Lukasiewicz's logic and their semiring reducts.Contemp. Math. 377 (2005), 131–144. Zbl 1081.06009, MR 2149001, 10.1090/conm/377/06988 |
| Reference:
|
[5] Di Nola A., Lettieri A.: Perfect MV-algebras are categorically equivalent to abelian $\ell$-groups.Studia Logica 53 (1994), 417–432. MR 1302453, 10.1007/BF01057937 |
| Reference:
|
[6] Golan J.S.: Semirings and Their Applications.Kluwer Academic, Dordrecht, 1999. Zbl 0947.16034, MR 1746739 |
| Reference:
|
[7] Ježek J., Kala V., Kepka T.: Finitely generated algebraic structures with various divisibility conditions.Forum Math. 24 (2012), 379–397. Zbl 1254.16041, MR 2900012, 10.1515/form.2011.068 |
| Reference:
|
[8] Kala V.: Lattice-ordered groups finitely generated as semirings.J. Commut. Alg., to appear, 16 pp., arxiv:1502.01651. MR 3685049 |
| Reference:
|
[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, MR 2432815 |
| Reference:
|
[10] Kala V., Kepka T., Korbelář M.: Notes on commutative parasemifields.Comment. Math. Univ. Carolin. 50 (2009), 521–533. Zbl 1203.16038, MR 2583130 |
| Reference:
|
[11] Kala V., Korbelář M.: Idempotence of commutative semifields.preprint, 16 pp. |
| Reference:
|
[12] Kepka T., Korbelář M.: Conjectures on additively divisible commutative semirings.Math. Slovaca 66 (2016), 1059–1064. MR 3602603, 10.1515/ms-2016-0203 |
| Reference:
|
[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, MR 3608425, 10.1142/S0219498817500384 |
| Reference:
|
[14] Korbelář M.: Torsion and divisibility in finitely generated commutative semirings.Semigroup Forum, to appear; DOI: 10.1007/s00233-016-9827-4. MR 3715842, 10.1007/s00233-016-9827-4 |
| Reference:
|
[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. MR 3694657, 10.1007/s00208-017-1527-1 |
| Reference:
|
[16] Litvinov G.L.: The Maslov dequantization, idempotent and tropical mathematics: a brief introduction.arXiv:math/0507014. Zbl 1102.46049, MR 2148995 |
| Reference:
|
[17] Monico C.J.: Semirings and semigroup actions in public-key cryptography.PhD Thesis, University of Notre Dame, USA, 2002. MR 2703068 |
| Reference:
|
[18] Mundici D.: Interpretation of AF $C^*$-algebras in Łukasiewicz sentential calculus.J. Funct. Anal. 65 (1986), 15–63. MR 0819173, 10.1016/0022-1236(86)90015-7 |
| Reference:
|
[19] Weinert H.J.: Über Halbringe und Halbkörper I.Acta Math. Acad. Sci. Hungar. 13 (1962), 365–378. Zbl 0125.01002, MR 0146108, 10.1007/BF02020799 |
| Reference:
|
[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, MR 1407915, 10.1007/BF01879738 |
| Reference:
|
[21] Zumbrägel J.: Public-key cryptography based on simple semirings.PhD Thesis, Universität Zürich, Switzerland, 2008. |
| . |
