Davenport constant

For a finite Abelian group , Davenport's constant of is the smallest positive integer such that for any sequence of not necessarily distinct elements of there is a non-empty subsequence with sum zero (i.e., there is some with ).

It is reported in [a7] that in 1966 H. Davenport proposed the problem of finding in the following connection. Let be an algebraic number field (cf. also Algebraic number; Field) with ring of integers and ideal class group . Then is the maximal number of prime ideals (counted with multiplicity) which can divide an irreducible element of . This is the reason why is a crucial invariant in the theory of non-unique factorizations [a2].

Let denote the cyclic group with elements and suppose with and with rank . Then

In the left-hand inequality, equality holds for -groups and for groups with (proved independently by J.E. Olson [a7] and D. Kruyswijk [a3]; cf. also -group). However, the left-hand inequality can be strict [a4], [a5]. For the right-hand inequality, cf. [a6]. It is still (1996) an open question whether the left-hand inequality can be strict for groups of rank or for groups of the form .

The problem of determining belongs to the area of zero-sum sequences, a subfield of additive number theory, respectively additive group theory. For related problems cf. [a1] and the literature cited there.

References

[a1]	N. Alon, M. Dubiner, "Zero-sum sets of prescribed size" D. Miklós (ed.) V.T. Sós (ed.) T. Szönyi (ed.) , Combinatorics, Paul Erdös is Eighty , Bolyai Society Mathematical Studies , 1 , Keszthely (Hungary) (1993) pp. 33–50
[a2]	S. Chapman, "On the Davenport's constant, the cross number and their application in factorization theory" , Lecture Notes in Pure and Appl. Math. , 171 , M. Dekker (1995) pp. 167–190
[a3]	P. van Emde Boas, D. Kruyswijk, "A combinatorial problem on finite abelian groups III" Report Math. Centre , ZW–1969–008 (1969)
[a4]	A. Geroldinger, R. Schneider, "On Davenport's constant" J. Combin. Th. A , 61 (1992) pp. 147–152
[a5]	M. Mazur, "A note on the growth of Davenport's constant" Manuscr. Math. , 74 (1992) pp. 229–235
[a6]	R. Meshulam, "An uncertainty inequality and zero subsums" Discrete Math. , 84 (1990) pp. 197–200
[a7]	J.E. Olson, "A combinatorial problem on finite abelian groups I–II" J. Number Th. , 1 (1969) pp. 8–10; 195–199