W.A. Schmid's homepage (Research)

Research (W. A. Schmid)

My research focuses on two related subjects.

Zero-sum problems in finite abelian groups

A classical problem, popularized by Davenport, is the following: Let (G,+) denote a finite abelian group. Determine the smallest D(G) such that each sequence in G of length at least D(G) has a zero-sum subsequence, that is a subsequence the sum of whose terms is the zero-element of G. And, determine the structure of extremal sequences without a zero-sum subsequence, that is solve the associated inverse problem. This and related problems have been investigated since the 1960s, using various tools from Additive Number Theory/Combinatorics. Still precise answers are only known in special cases.
Some years ago inverse problems for groups of rank two and more generally groups with "large" exponent in a relative sense (answers for cyclic groups are well-known), were a main focus of my research. More recently, I worked mainly on "weighted" problems, but I still have some projects related to the former, too.

Non-unique factorization theory

The maximal order of a number field (and more generally a Dedekind domain or Krull monoid) is factorial if and only if its ideal class group is trivial. Thus, if the class group is non-trivial then there exist elements that do not have a unique factorization into irreducible elements, but several factorizations that are distinct (in an essential way).
A goal of non-unique factorization theory is to understand the various phenomena of non-uniqueness that can arise for non-trivial class group both from a qualitative and quantitative point of view. In other words, one wishes to give an explicit meaning to: "the class group determines the arithmetic."
Trying to answer these questions, leads to various kinds of zero-sum problems in the class group, thus establishing the link between these two subjects.
Commutative Algebra, specifically Multiplicative Ideal Theory, and (abstract) Analytic Number Theory are two other key tools.

More recently, my work also explores connections of the subjects mentioned above to Coding Theory.

Publications and preprints

The list below is not always up to date. However, my preprints appear rapidly upon completion on HAL and arXiv.
(The preprints possibly differ slightly from the published ones.)

Title	Coauthor(s)	Bibliographical information	local copies / link to published version / arXiv or HAL
Which sets are sets of lengths in all numerical monoids?	A. Geroldinger	submitted	https://hal.archives-ouvertes.fr/hal-01700170
On congruence half-factorial Krull monoids with cyclic class group	A. Plagne	submitted	https://hal.archives-ouvertes.fr/hal-01580928
Systems of sets of lengths: transfer Krull monoids vs. weakly Krull monoids	A. Geroldinger, Q. Zhong	In: "Rings, Polynomials, and Modules" Springer 2017. Pages 191--235.	https://hal.archives-ouvertes.fr/hal-01332417
A realization theorem for sets of lengths in numerical monoids	A. Geroldinger	Forum Math., to appear	https://hal.archives-ouvertes.fr/hal-01615120
A realization theorem for sets of distances	A. Geroldinger	J. Algebra 481, 188--198, 2017	https://hal.archives-ouvertes.fr/hal-01355253
A characterization of class groups via sets of lengths	A. Geroldinger	submitted	https://hal.archives-ouvertes.fr/hal-01131955
Some recent results and open problems on sets of lengths of Krull monoids with finite class group		In: "Multiplicative ideal theory and factorization theory" Springer, Proc. Math. Stat. 170, 2016. Pages 323--352.	https://hal.archives-ouvertes.fr/hal-01233263
Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory	L. E. Marchan, O. Ordaz, I. Santos	J. Combin. Theory Ser. A 135, 237--267, 2015	http://hal.archives-ouvertes.fr/hal-01020156
The system of sets of lengths in Krull monoids under set addition	A. Geroldinger	Rev. Mat. Iberoam. 32, 571--588, 2016	http://hal.archives-ouvertes.fr/hal-01020157
Local and global tameness in Krull monoids	W. D. Gao, A. Geroldinger	Comm. Algebra 43, 262--296, 2015.	http://hal.archives-ouvertes.fr/hal-00787681
Remarks on the plus-minus weighted Davenport constant	L. E. Marchan, O. Ordaz	Int. J. Number Theory 10, 1219--1239, 2014	http://hal.archives-ouvertes.fr/hal-00835688
Some exact values of the Harborth constant and its plus-minus weighted analogue	L. E. Marchan, O. Ordaz, D. Ramos	Arch. Math. (Basel) 101, 501--512, 2013	http://hal.archives-ouvertes.fr/hal-00851556
Some remarks on barycentric-sum problems over cyclic group	A. Plagne, O.Ordaz	European J. Combin. 34, 1415--1428, 2013	http://hal.archives-ouvertes.fr/hal-00689464
On the Olson and the Strong Davenport constants	A. Philipp, O. Ordaz, I. Santos	J. Theor. Nombres Bordeaux, 23, 715--750, 2011.	pdf
An application of coding theory to estimating Davenports constants	A. Plagne	Des. Codes Cryptogr., 61, 105--118, 2011.	pdf
Zero-sum problems with congruence conditions	A. Geroldinger, D.J.Grynkiewicz	Acta Math. Hungar., 131, 323--345, 2011.	pdf
Restricted inverse zero-sum problems in groups of rank two.		Q. J. Math., 63, 477--487, 2012.	pdf
The catenary degree of Krull monoids I	A.Geroldinger, D.J.Grynkiewicz	J. Th\'eor. Nombres Bordeaux, 23, 137--169, 2011.	pdf / arXiv:0911.4882
On the arithmetic of Krull monoids with infinite cyclic class group	A.Geroldinger, D.J.Grynkiewicz, G.J.Schaeffer	J. Pure Appl. Alg., 214:2219--2250, 2010	pdf / arXiv:0908.4191
Remarks on a generalization of the Davenport constant	M. Freeze	Discrete Math., 310:3373--3389, 2010	pdf / arXiv:0905.4248
The inverse problem associated to the Davenport constant for C_2+C_2+C_2n, and applications to the arithmetical characterization of class groups		Electron. J. Combin., 18, P33, 42pp, 2011.	pdf
Higher-order class groups and block monoids of Krull monoids with torsion class group		J. Algebra Appl., 9:433--464, 2010	pdf
Inverse zero-sum problems II		Acta Arith., 143:333--343, 2010	pdf / arXiv:0801.3747
Characterization of class groups of Krull monoids via their systems of sets of lengths: a status report		In: "Number Theory and Applications: Proceedings of the International Conferences on Number Theory and Cryptography (Eds. S.D.Adhikari and B.Ramakrishnan, HRI)", Hindustan Book Agency, 2009. Pages 189--212.	pdf
Arithmetical characterization of class groups of the form $Z/n Z \oplus Z/nZ$ via the system of sets of lengths		Abh. Math. Sem. Hamburg, 79:25--35, 2009	pdf / SpringerLink: doi:10.1007/s12188-008-0010-z
A realization theorem for sets of lengths		J. Number Theory, 129:990--999, 2009	pdf / ScienceDirect (Elsevier): doi:10.1016/j.jnt.2008.10.019
On minimal distances in Krull monoids with infinite class group	S.T.Chapman, W.W.Smith	Bull. London Math. Soc., 40:613--618, 2008	pdf / Free access link to final version at oxfordjournals.org
Periods of sets of lengths: a quantitative result and an associated inverse problem		Colloq. Math., 113:33--53, 2008	pdf / Colloquium Mathematicum
Quasi half-factorial subsets of abelian torsion groups	S.T.Chapman, W.W.Smith	Houston J. Math., 35:13--22, 2009	pdf
On short zero-sum subsequences over p-groups	J.J.Zhuang	Ars Combin., 95:343--352, 2010	pdf
Inverse zero-sum problems	W.D.Gao, A.Geroldinger	Acta Arith., 128:245-279, 2007	pdf / Acta Arithmetica
On short zero-sum subsequences II	W.D.Gao, Q.H.Hou, R.Thangadurai	Integers, 7: A21, 22 pp. (electronic), 2007	pdf / INTEGERS (a free journal)
Weakly half-factorial sets in finite abelian groups	M.Radziejewski	Forum Math., 19:727--747, 2007	pdf / atypon-link (Walter de Gruyter): doi:10.1515/FORUM.2007.028
Minimal zero-sum sequences in $C_n\oplus C_n$	G.Lettl	European J. Combin., 28:742--753, 2007.	pdf / ScienceDirect (Elsevier): doi:10.1016/j.ejc.2005.12.008
On the set of integral solutions of the Pell equation in number fields		Aequationes Math., 71:109--114, 2006	pdf / SpringerLink: doi:10.1007/s00010-005-2782-x
Half-factorial sets in elementary $p$-groups		Far East J. Math. Sci. (FJMS), 22:75--114, 2006	pdf
Half-factorial sets in finite abelian groups: a survey		Grazer Math. Ber., 348:41--64, 2005	pdf
On the maximal cardinality of half-factorial sets in cyclic groups	A.Plagne	Math. Ann., 333:759--785, 2005	pdf / SpringerLink: doi:10.1007/s00208-005-0690-y
On invariants related to non-unique factorizations in block monoids and rings of integers		Math. Slovaca, 55:21--37, 2005	pdf
On large half-factorial sets in elementary $p$-groups: Maximal cardinality and structural characterization	A.Plagne	Israel J. Math., 145:285--310, 2005	pdf
On the asymptotic behavior of some counting functions, II		Colloq. Math., 102:197--216, 2005.	pdf / Colloquium Mathematicum
On the asymptotic behavior of some counting functions	M.Radziejewski	Colloq. Math., 102:181--195, 2005.	pdf / Colloquium Mathematicum
On zero-sum subsequences of restricted size IV	R.Chi, S.Ding, W.Gao, A.Geroldinger	Acta. Math. Hungar., 107:337--344, 2005.	pdf / SpringerLink: doi:10.1007/s10474-005-0201-3
Differences in sets of lengths of Krull monoids with finite class group		J. Th\'eor. Nombres Bordeaux, 17:323--345, 2005.	pdf / Journal de Th\'eorie des Nombres de Bordeaux
Arithmetic of block monoids		Math. Slovaca, 54:503--526, 2004	pdf
On the dual of an ideal and ideal transforms	A.Foroutan, F.Halter-Koch	Bull. Greek Math. Soc., 48:19--40, 2003.	pdf (scan)
On zero-sum subsequences in finite abelian groups		Integers, 1:A1, 8 pp. (electronic), 2001.	pdf / INTEGERS (a free journal)

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last changed: February 2018 by schmid"at"math.univ-paris13.fr