We give the maximal length of a Newton or a Schinzel sequence in a quadratic extension of a global field. In the case of a number field, the maximal length of a Schinzel sequence is 1, except in seven particular cases, and the Newton sequences are also finite, except for at most finitely many cases, all real. We give the maximal length of these sequences in the special cases. We have similar results in the case of a quadratic extension of a function field , taking in account that the ring of integers may be isomorphic to , in which case there are obviously infinite Newton and Schinzel sequences.
DOI : 10.5802/acirm.28
Mots clés : Integer-valued polynomials, Newton and Schinzel sequences, Quadratic number and function fields
@article{ACIRM_2010__2_2_15_0,
author = {Adam, David and Cahen, Paul-Jean},
title = {Newton and {Schinzel} sequences in quadratic fields},
journal = {Actes des rencontres du CIRM},
pages = {15--20},
publisher = {CIRM},
volume = {2},
number = {2},
year = {2010},
doi = {10.5802/acirm.28},
zbl = {06938576},
language = {en},
url = {http://www.numdam.org/articles/10.5802/acirm.28/}
}
TY - JOUR AU - Adam, David AU - Cahen, Paul-Jean TI - Newton and Schinzel sequences in quadratic fields JO - Actes des rencontres du CIRM PY - 2010 SP - 15 EP - 20 VL - 2 IS - 2 PB - CIRM UR - http://www.numdam.org/articles/10.5802/acirm.28/ DO - 10.5802/acirm.28 LA - en ID - ACIRM_2010__2_2_15_0 ER -
Adam, David; Cahen, Paul-Jean. Newton and Schinzel sequences in quadratic fields. Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 15-20. doi : 10.5802/acirm.28. http://www.numdam.org/articles/10.5802/acirm.28/
[1] D. Adam, Simultaneous orderings in function fields, J. Number Theory 112 (2005), 287–297. | DOI | MR | Zbl
[2] —, Pólya and Newtonian function fields, Manuscripta Math. 126 (2008), no. 2, 231–246. | DOI | Zbl
[3] Y. Amice, Interpolation -adique, Bull. Soc. Math. France 92 (1964), 117–180. | DOI | Zbl
[4] M. Bhargava, -orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math. 490 (1997), 101–127. | DOI | MR | Zbl
[5] —, The factorial function and generalizations, Amer. Math. Monthly 107 (2000), 783–799. | DOI | MR | Zbl
[6] P.J. Cahen, Newtonian and Schinzel sequences in a domain, J. of Pure and Appl. Algebra 213 (2009), 2117–2133. | DOI | MR | Zbl
[7] P.J. Cahen, J.L. Chabert, Integer valued polynomials, Mathematical Survey and Monographs,vol 48, American Mathematical Society, Providence, (1997) | Zbl
[8] —, Old Problems and New Questions around Integer-Valued Polynomials and Factorial Sequences, Multiplicative ideal theory in commutative algebra, Springer, New York, (2006), 89–108. | DOI
[9] M. Car, Répartition modulo dans un corps de séries formelles sur un corps fini, Acta Arith. 69.3 (1995), 229–242. | DOI | Zbl
[10] A. Granville, R.A. Mollin, H.C. Williams, An upper bound on the least inert prime in a real quadratic field, Canad. J. Math. 52.2 (2000), 369–380. | DOI | MR | Zbl
[11] J. Latham, On sequences of algebraic integers, J. London Math. Soc. 6.2 (1973), 555–560. | DOI | Zbl
[12] PARI/GP, version 2.3.4, Bordeaux, 2008, http://pari.math.u-bordeaux.fr. | DOI | MR
[13] W. Narkiewicz, Some unsolved problems, Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), Bull. Soc. Math. France, Mem. 25 (1971), 12–02. | Numdam
[14] B. Wantula, Browkin’s problem for quadratic fields. (Polish) Zeszyty Nauk. Politech. Ślpolhk ask. Mat.-Fiz. 24 (1974), 173–178. | Zbl
[15] R. Wasen, On sequences of algebraic integers in pure extensions of prime degree, Colloq. Math. 30 (1974), 89–104. | DOI | MR | Zbl
[16] M. Wood, -orderings: a metric viewpoint and the non-existence of simultaneous orderings, J. Number Theory 99 (2003), 36–56. | DOI | MR | Zbl
[17] J. Yéramian, Anneaux de Bhargava, Comm. Algebra 32.8, (2004), 3043–3069. | DOI | MR | Zbl
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