An overview of some recent developments on integer-valued polynomials: Answers and Questions
- [1] Départment de Mathématiques Université de Picardie 80039 Amiens, France LAMFA CNRS-UMR 6140
Actes des rencontres du CIRM (2010)
- Volume: 2, Issue: 2, page 3-14
- ISSN: 2105-0597
Access Full Article
topAbstract
topHow to cite
topChabert, Jean-Luc. "An overview of some recent developments on integer-valued polynomials: Answers and Questions." Actes des rencontres du CIRM 2.2 (2010): 3-14. <http://eudml.org/doc/196271>.
@article{Chabert2010,
abstract = {The purpose of my talk is to give an overview of some more or less recent developments on integer-valued polynomials and, doing so, to emphasize that integer-valued polynomials really occur in different areas: combinatorics, arithmetic, number theory, commutative and non-commutative algebra, topology, ultrametric analysis, and dynamics. I will show that several answers were given to open problems, and I will raise also some new questions.},
affiliation = {Départment de Mathématiques Université de Picardie 80039 Amiens, France LAMFA CNRS-UMR 6140},
author = {Chabert, Jean-Luc},
journal = {Actes des rencontres du CIRM},
keywords = {Integer-valued polynomials; Pólya fields; Prüfer domains; $p$-orderings; Quaternions; Bernoulli numbers; ultrametric analysis; discrete dynamical systems},
language = {eng},
number = {2},
pages = {3-14},
publisher = {CIRM},
title = {An overview of some recent developments on integer-valued polynomials: Answers and Questions},
url = {http://eudml.org/doc/196271},
volume = {2},
year = {2010},
}
TY - JOUR
AU - Chabert, Jean-Luc
TI - An overview of some recent developments on integer-valued polynomials: Answers and Questions
JO - Actes des rencontres du CIRM
PY - 2010
PB - CIRM
VL - 2
IS - 2
SP - 3
EP - 14
AB - The purpose of my talk is to give an overview of some more or less recent developments on integer-valued polynomials and, doing so, to emphasize that integer-valued polynomials really occur in different areas: combinatorics, arithmetic, number theory, commutative and non-commutative algebra, topology, ultrametric analysis, and dynamics. I will show that several answers were given to open problems, and I will raise also some new questions.
LA - eng
KW - Integer-valued polynomials; Pólya fields; Prüfer domains; $p$-orderings; Quaternions; Bernoulli numbers; ultrametric analysis; discrete dynamical systems
UR - http://eudml.org/doc/196271
ER -
References
top- M. Ably, Fonctions entières à valeurs dans un corps de nombres, Bull. Soc. Math. France139 (2011), 243–270. Zbl1219.11045
- D. Adam, Car-Pólya and Gel’fond’s theorems for , Acta Arith.115 (2004), 287-303. Zbl1088.11087MR2100507
- D. Adam, Simultaneous orderings in function fields, J. Number Theory112 (2005), 287–297. Zbl1075.11075MR2141533
- D. Adam, Pólya and Newtonian function fields, Manuscripta Math.126 (2008), 231–246. Zbl1141.11057MR2403187
- D. Adam, J.-L. Chabert, and Y. Fares, Subsets of with simultaneous orderings, Integers10 (2010), 435–451. Zbl1246.11074MR2798615
- D. Adam and N. Hirata-Kohno, Almost integer-valued functions in positive characteristic, Diophantine analysis and related fields, 2006, 1–10, Sem. Math. Sci., 35, Keio Univ., Yokohama, 2006. Zbl1148.13301MR2331644
- Y. Amice, Interpolation -adique, Bull. Soc. Math. France92 (1964) 117–180. Zbl0158.30201MR188199
- D. Barsky, Fonctions -lipschitziennes sur un anneau local et polynômes à valeurs entières, Bull. Soc. Math. France101 (1973), 397–411. Zbl0291.12107MR371863
- M. Bhargava, -orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math.490 (1997), 101–127. Zbl0899.13022MR1468927
- M. Bhargava, Generalized Factorials and Fixed Divisors over Subsets of a Dedekind Domain, J. Number Theory72 (1998), 67–75. Zbl0931.13004MR1643292
- M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly107 (2000), 783–799. Zbl0987.05003MR1792411
- M. Bhargava, On -orderings, rings of integer-valued polynomials, and ultrametric analysis, J. Amer. Math. Soc.22 (2009), 963Ð993. Zbl1219.11047MR2525777
- M. Bhargava, P.-J. Cahen, and J. Yeramian, Finite generation properties for rings of integer-valued polynomials, J. Algebra322 (2009), 1129–1150. Zbl1177.13051MR2537676
- M. Bhargava and K. S. Kedlaya, Continuous functions on compact subsets of local fields, Acta Arith.1999, 191–198. Zbl0979.11054MR1735672
- J. Boulanger and J.-L. Chabert, Asymptotic behavior of characteristic sequences of integer-valued polynomials, J. Number Theory80 (2000), 238–259. Zbl0973.11093MR1740513
- J. Brewer and L. Klinger, Rings of Integer-Valued Polynomials and the bcs-Property, Commutative Ring Theory and Applications, 65–75, Lecture Notes in Pure and Appl. Math. 231, Dekker, New York, 2003. Zbl1080.13508MR2029818
- D. Brizolis, A theorem on ideals in Prüfer rings of integral-valued polynomials, Comm. Algebra7 (1979), 1065–1077. Zbl0422.13011MR533204
- P.-J.Cahen, Dimension de l’anneau des polynômes à valeurs entières, Manuscripta Math., 67 (1990), 333–343. Zbl0712.13001MR1046992
- P.-J. Cahen, Polynomial closure, J. Number Theory61 (1996), 226–247. Zbl0877.13007MR1423051
- P.-J. Cahen, Newtonian and Schinzel sequences in a domain, J. Pure Appl. Algebra213 (2009), 2117–2133. Zbl1170.13007MR2533310
- P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials, Amer. Math. Soc. Surveys and Monographs 48, Povidence, R.I., 1997. Zbl0884.13010MR1421321
- P.-J. Cahen and J.-L. Chabert, On the ultrametric Stone-Weierstrass theorem and Mahler’s expansion , J. Théorie des Nombres Bordeaux, 14 (2002), 43-57. Zbl1031.46086MR1925989
- P.-J. Cahen, J.-L. Chabert, and K.A. Loper, High dimension Prüfer domains of integer-valued polynomials, J. Korean Math. Soc., 38 (2001), 915–935. Zbl1010.13011MR1849332
- M. Car, Pólya’s theorem for , Acta Arith.69 (1995), 229–242.
- M. Car, Entire functions and integer-valued polynomials in positive characteristic, Third International Meeting on Integer-Valued Polynomials, C.I.R.M., Marseille, Nov.-Dec. 2010.
- J.-L. Chabert, Anneaux de polynômes à valeurs entières et anneaux de Fatou, Bull. Soc. math. France99 (1971), 273–283 . Zbl0202.32802MR302636
- J.-L. Chabert, Integer-valued polynomials on prime numbers and logarithm power expansion, European J. Combinatorics28 (2007), 754–761. Zbl1120.11016MR2300756
- J.-L. Chabert, On the polynomial closure in a valued field, J. Number Theory130 (2010), 458–468. Zbl1245.13014MR2564907
- J.-L. Chabert, Does have the stacked basis property?, Arabian J. Math., to appear.
- J.-L. Chabert, Dynamical systems and -orderings, preprint.
- J.-L. Chabert and P.-J. Cahen, Old Problems and New Questions around Integer-Valued Polynomials and Factorial Sequences, in Multiplicative Ideal Theory in Commutative Algebra, Springer 2006, 89-108. Zbl1117.13022MR2265803
- J.-L. Chabert, S. Chapman and W. Smith, A Basis for the Ring of Polynomials Integer-Valued on Prime Numbers, in Factorization in Integral Domains, Lecture Notes in Pure and Appl. Math., 189, Dekker, New York, 1997, 271–284. Zbl0967.13015MR1460778
- J.-L. Chabert, S. Evrard, and Y. Fares, Regular subsets of valued fields and Bhargava’s -orderings, preprint. Zbl1270.13015
- J.-L. Chabert, A.-H. Fan, and Y. Fares , Minimal dynamical systems on a discrete valuation domain, Discrete Contin. Dyn. Syst.35 (2009), 777–795. Zbl1185.37014MR2533975
- L. Delamette, Théorème de Pólya en caractéristique finie, Acta Arith.106 (2003), 159–170. Zbl1054.11037MR1958982
- S. Evrard and Y. Fares, -adic subsets whose factorials satisfy a generalized Legendre formula, Bull. London Math. Soc., 40 (2008), 37-50. Zbl1160.11011MR2409176
- M. Fontana, L. Izelgue, S. Kabbaj, and F.Tartarone, On the Krull dimension of domains of integer-valued polynomials, Expo. Math., 15 (1997) 433–465. Zbl0920.13005MR1486408
- M. Fontana and S. Kabbaj, Essential domains and two conjectures in dimension theory, Proc. Amer. math. Soc., 123 (2004), 2529–2535. Zbl1059.13008MR2054776
- S. Frisch, Substitution and closure of sets under integer-valued polynomials, J. Number Theory56 (1996), 396–403. Zbl0844.13014MR1373562
- S. Frisch, Polynomial separation of points in algebras, in Arithmetical Properties of Commutative Rings and Monoids, S. Chapman ed., Dekker, 2005, 249–254. Zbl1092.13027MR2140698
- L. Fuchs and L. Salce, Modules over Non-Noetherian Domains, Amer. Math. Soc. Surveys and Monographs 84. A.M.S., Providence, 2001. Zbl0973.13001MR1794715
- R. Gilmer, Sets that determine integer-valued polynomials, J. Number Theory33 (1989), 95–100. Zbl0695.13015MR1014390
- F. Gramain, Sur le théorème de Fukusawa-Gel’fond, Invent. Math.63 (1981), 495–506. Zbl0461.10028MR620681
- K. Johnson, The invariant subalgebra and anti-invariant submodule of , J. K-theory2 (2008), 123–145. Zbl1178.55014MR2434169
- K. Johnson, -orderings of Finite Subsets of Dedekind Domains, J. Algebraic Combinatorics30 (2009), 233–255. Zbl1180.13024MR2525060
- K. Johnson, Limits of characteristic sequences of integer-valued polynomials on homogeneous sets, J. Number Theory129 (2009), 2933–2942. Zbl1233.11030MR2560844
- I. Kaplansky, The Weierstrass theoremin fields with valuations, Proc. Amer. Math. Soc.1 (1950), 356–357. Zbl0038.07002MR35760
- A. Leriche, J. Théor. Nombres Bordeaux23 (2011), 235-249. MR2780627
- A. Leriche, Pólya fields, Pólya groups and PÊólya extensions: a question of capitulation, Ph.D. Thesis, Université de Picardie, Dec. 2010. Zbl1282.13040
- K. Mahler, An Interpolation Series for Continuous Functions of a -adic Variable, J. Reine Angew. Math.199 (1958), 23–34 and 208 (1961), 70–72. Zbl0080.03504MR95821
- A. Mingarelli, Abstract Factorials of arbitrary Sets of Integers, arXiv: 0705.4299v1, 29 may 2007.
- H. Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math.101 (1897), 196–202.
- A. Ostrowski, Ueber ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math.149 (1919), 117–124. Zbl47.0163.05
- A. Ostrowski, Untersuchungen zur arithmetischen Theorie der Körper, Math. Zeitschrift39 (1935), 269–404. Zbl0010.15001
- M.H. Park, Polynomial closure in essential domains, Manuscripta Math.117 (2005), 29–41. Zbl1094.13033MR2142899
- G. Pólya, Ueber ganzwertige ganze Funktionen, Rend. Circ. Matem. Palermo40 (1915), 1–16. Zbl45.0655.02
- G. Pólya, Ueber ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math.149 (1919), 97–116. Zbl47.0163.04
- I. Schur, Über eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1905), 77–91.
- M.-J. Strong and S. Whitehouse, Integer-valued polynomials and K-theory operations, Proc. Amer. Math. Soc.138 (2010), 2221-2233. Zbl1195.55013MR2596063
- F. Tartarone, Polynomial closure in Noetherian pseudo-valuation domains, Comm. Algebra31 (2003), 5431–5446. Zbl1039.13014MR2005236
- F. J. Van Der Linden, Integer valued polynomials over function fields, Nederl. Akad. Wetensch. Indag. Math.50 (1988), 293–308. Zbl0674.14014MR964835
- N. Werner, Integer-Valued Polynomials over Quaternions Rings, J. Algebra324 (2010), 1754–1769. Zbl1219.16027MR2673759
- M. Wood, -orderings: a metric viewpoint and the non-existence of simultaneous orderings, J. Number Theory99 (2003), 36–56. Zbl1076.13011MR1957243
- J. Yeramian, Anneaux de Bhargava, Comm. in Algebra32 (2004), 3043-3069. Zbl1061.13011MR2102166
- J. Yeramian, Anneaux de Bhargava, Ph.D. Thesis, Marseille, 2004. MR2102166
- H. Zantema, Integer valued polynomials over a number field, Manuscr. Math.40 (1982), 155–203. Zbl0505.12003MR683038
- 1990: Integer-Valued Polynomial Encounter, CIRM 11-12 December 1990, Laboratoire de Mathématiques URA CNRS 225, Marseille, Prépublication n 91-17.
- 2000: http://www.cirm.univ-mrs.fr/liste_rencontre/archives/Rencontres2000/Cahen2000.html
- 2010: http://www.lamfa.u-picardie.fr/evrard/colloqueIVP/index.htm
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.