Combinatorical and Algebraic Structures Seminar

Session details

Date: 27.3.2012
Speaker: Daniel Dombek, FJFI, České vysoké učení technické v Praze
Title: On the generalizations of the unit sum number problem
Abstract: Consider a ring of integers $O_K$ in an algebraic number field $K$. The so-called unit sum number problem asks a question: ``Can we express any algebraic integer in $K$ as sum of units? And if we can, is there a bound for the number of summands?'' In this talk we will recall some known results concerning sums of units. Furthermore, we will show that the unit sum number problem can be generalized. In particular, instead of sums of units we consider sums of elements of $O_K$ with bounded norm, or even more general setting -- linear combinations of units. References: A. Bérczes, L. Hajdu, A. Peth\H o: ``Arithmetic progressions in the solution sets of norm form equations''. Rocky Mountain Math. J. 40 (2010), 383--396. D. Dombek, L. Hajdu, A. Peth\H o: ``Representing algebraic integers as linear combinations of units''. Preprint (2012), 10pp. L. Hajdu: ``Arithmetic progressions in linear combinations of S-units''. Period. Math. Hungar. 54 (2007), 175--181. M. Jarden, W. Narkiewicz: ``On Sums of Units''. Monatsh. Math. 150 (2007), 327--332.

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