×

Approximation for Frobenius algebraic equations in Witt vectors. (English) Zbl 1166.13030

Summary: We prove an approximation property for solutions to difference equations in excellent discrete valuation rings satisfying an appropriate Hensel’s lemma, analog to a theorem of M. J. Greenberg [Publ. Math., Inst. Hautes Étud. Sci. 31, 563–568 (1966; Zbl 0146.42201)]. In the case of Witt vectors we obtain a Nullstellensatz for Frobenius algebraic equations.

MSC:

13K05 Witt vectors and related rings (MSC2000)

Citations:

Zbl 0146.42201
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Becker, J.; Denef, J.; Lipshitz, L.; van den Dries, L., Ultraproducts and approximation in local rings I, Invent. Math., 51, 189-203 (1979) · Zbl 0416.13004
[2] Bélair, L.; Macintyre, A., L’automorphisme de Frobenius des vecteurs de Witt, C. R. Acad. Sci. Paris Sér. I, 331, 1-4 (2000) · Zbl 0966.03038
[3] Bélair, L., Fonctions rationnelles aux différences à valeurs entières dans les vecteurs de Witt, C. R. Acad. Sci. Paris Sér. I, 339, 83-86 (2004) · Zbl 1050.03027
[4] Bélair, L., Équations aux différences dans les vecteurs de Witt, C. R. Acad. Sci. Paris Sér. I, 340, 99-102 (2005) · Zbl 1107.13029
[5] Bélair, L.; Macintyre, A.; Scanlon, T., Model theory of the Frobenius on the Witt vectors, Amer. J. Math., 129, 665-721 (2007) · Zbl 1121.03043
[6] Buium, A., An approximation property for Teichmüller points, Math. Res. Lett., 3, 453-457 (1996) · Zbl 0902.14015
[7] Chatzidakis, Z., Generic automorphisms of separably closed fields, Illinois J. Math., 45, 693-733 (2001) · Zbl 0993.03047
[8] Cohn, R. M., Difference Algebra (1965), Wiley · Zbl 0127.26402
[9] L. van den Dries, Model theory of fields, decidability, and bounds for polynomial ideals, Doctoral thesis, Rijkuniversiteit te Utrecht, Netherlands, 1978; L. van den Dries, Model theory of fields, decidability, and bounds for polynomial ideals, Doctoral thesis, Rijkuniversiteit te Utrecht, Netherlands, 1978
[10] Duval, A., Lemmes de Hensel et factorisation formelle pour les opérateurs aux différences, Funkcial. Ekvac., 26, 349-368 (1983) · Zbl 0543.12018
[11] Dwork, B.; Robba, P., On ordinary linear \(p\)-adic differential equations, Trans. Amer. Math. Soc., 231, 1-46 (1977) · Zbl 0375.34010
[12] Greenberg, M., Rational points in henselian discrete valuation rings, Publ. Math. Inst. Hautes Études Sci., 31, 59-64 (1966) · Zbl 0142.00901
[13] Greenberg, M., Strictly local solutions of diophantine equations, Pacific J. Math., 51, 143-153 (1974) · Zbl 0247.10015
[14] Guzy, N., Quelques remarques sur les corps \(D\)-valués, C. R. Acad. Sci. Paris Sér. I, 343, 689-694 (2006) · Zbl 1108.03041
[15] D. Haskell, Y. Yaffe, Ganzestellensätze in theories of valued fields, preprint; D. Haskell, Y. Yaffe, Ganzestellensätze in theories of valued fields, preprint · Zbl 1232.12007
[16] Jacobson, N., Basic Algebra II (1987), Freeman
[17] Joyal, A., \(δ\)-anneaux et vecteurs de Witt, C.R. Math. - Math. Rep. Acad. Sci. Canada, 7, 177-182 (1985) · Zbl 0594.13023
[18] Kolchin, E. R., Differential Algebra and Algebraic Groups (1973), Academic Press · Zbl 0264.12102
[19] Kochen, S., Integral valued rational functions over \(p\)-adic numbers: A \(p\)-adic analogue of the theory of real fields, (LeVeque, W. J.; Strauss, E. G., Proc. Sympos. Pure Math., vol. XII (1969), Amer. Math. Soc.), 57-73
[20] Marker, D., Model Theory: An Introduction (2002), Springer · Zbl 1003.03034
[21] Popescu, D., Artin approximation, (Hazewinkel, M., Handbook of Algebra, vol. 2 (2003), Elsevier), 321-356 · Zbl 1005.13003
[22] Prestel, A.; Roquette, P., Formally \(p\)-Adic Fields (1984), Springer · Zbl 0523.12016
[23] Robinson, A., Elementary embeddings of fields of power series, J. Number Theory, 2, 237-247 (1970) · Zbl 0199.30403
[24] Scanlon, T., A model-complete theory of valued \(D\)-fields, J. Symbolic Logic, 65, 1758-1784 (2000) · Zbl 0977.03021
[25] Scanlon, T., Diophantine geometry from model theory, Bull. Symbolic Logic, 7, 37-57 (2001) · Zbl 0984.03035
[26] Scanlon, T., Quantifier elimination for the relative Frobenius, (Kuhlmann, F. V.; etal., Valuation Theory and Its Applications, vol. II. Valuation Theory and Its Applications, vol. II, Saskatoon, SK, 1999 (2003), Amer. Math. Soc.), 323-352 · Zbl 1040.03031
[27] Seidenberg, A., Some basic theorems in differential algebra (characteristic \(p\), arbitrary), Trans. Amer. Math. Soc., 73, 174-190 (1952) · Zbl 0047.03502
[28] Serre, J.-P., Local Fields (1979), Springer
[29] Srhir, A., \(P\)-adic ideals of \(p\)-rank \(d\) and the \(p\)-adic Nullstellensatz, J. Pure Appl. Algebra, 180, 299-311 (2003) · Zbl 1101.13032
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.