Abstract
Let R be the ring of S-integers of an algebraic function field (in one variable) over a perfect field, where S is finite and not empty. It is shown that for every positive integer N there exist elements of R that can not be written as a sum of at most N units. Moreover, all quadratic global function fields whose rings of integers are generated by their units are determined.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ashrafi N., Vámos P.: On the unit sum number of some rings. Q. J. Math. 56(1), 1–12 (2005)
Belcher P.: Integers expressible as sums of distinct units. Bull. Lond. Math. Soc. 6, 66–68 (1974)
Belcher, P.: A test for integers being sums of distinct units applied to cubic fields. J. Lond. Math. Soc. (2) 12(2), 141–148 (1975/1976)
Brownawell W.D., Masser D.W.: Vanishing sums in function fields. Math. Proc. Camb. Philos. Soc. 100(3), 427–434 (1986)
Filipin A., Tichy R.F., Ziegler V.: The additive unit structure of pure quartic complex fields. Funct. Approx. Comment. Math. 39(1), 113–131 (2008)
Filipin A., Tichy R.F., Ziegler V.: On the quantitative unit sum number problem—an application of the subspace theorem. Acta Arith. 133(4), 297–308 (2008)
Fuchs C., Tichy R.F., Ziegler V.: On quantitative aspects of the unit sum number problem. Arch. Math. 93, 259–268 (2009)
Goldsmith B., Pabst S., Scott A.: Unit sum numbers of rings and modules. Q. J. Math. 49(195), 331–344 (1998)
Hajdu L.: Arithmetic progressions in linear combinations of S-units. Period. Math. Hung. 54(2), 175–181 (2007)
Hasse H.: Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. reine angew. Math. 172, 37–54 (1935)
Jacobson B.: Sums of distinct divisors and sums of distinct units. Proc. Am. Math. Soc. 15, 179–183 (1964)
Jarden M., Narkiewicz W.: On sums of units. Monatsh. Math. 150(4), 327–332 (2007)
Le Brigand, D.: Real quadratic extensions of the rational function field in characteristic two. In: Arithmetic, Geometry and Coding Theory (AGCT 2003), Sémin. Congr., vol. 11, pp. 143–169. Soc. Math. France, Paris (2005)
Mason R.C.: Norm form equations. I. J. Number Theory 22(2), 190–207 (1986)
Mason R.C.: Norm form equations. III. Positive characteristic. Math. Proc. Camb. Philos. Soc. 99(3), 409–423 (1986)
Rosen M.: Number Theory in Function Fields, Graduate Texts in Mathematics. Springer, New York (2002)
Śliwa J.: Sums of distinct units. Bull. Acad. Pol. Sci. 22, 11–13 (1974)
Stichtenoth H.: Algebraic Function Fields and Codes. Universitext. Springer, Berlin (1993)
Tichy R.F., Ziegler V.: Units generating the ring of integers of complex cubic fields. Colloq. Math. 109(1), 71–83 (2007)
Zelinsky D.: Every linear transformation is a sum of nonsingular ones. Proc. Am. Math. Soc. 5, 627–630 (1954)
Ziegler V.: The additive unit structure of complex biquadratic fields. Glas. Mat. 43(63)(2), 293–307 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by U. Zannier.
C. Frei is supported by the Austrian Science Foundation (FWF) project S9611-N23.
Rights and permissions
About this article
Cite this article
Frei, C. Sums of units in function fields. Monatsh Math 164, 39–54 (2011). https://doi.org/10.1007/s00605-010-0219-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-010-0219-7