5,663 research outputs found

    Minimal zero-sequences and the strong Davenport constant

    Get PDF
    AbstractLet G be a finite Abelian group and U(G) the set of minimal zero-sequences on G. If M1 and M2∈U(G), then set M1∼M2 if there exists an automorphism ϕ of G such that ϕ(M1)=M2. Let O(M) represent the equivalence class of M under ∼. In this paper, we consider problems related to the size of an equivalence class of sequences in U(G) and also examine a stronger form of the Davenport constant of G

    On the Olson and the Strong Davenport constants

    Get PDF
    A subset SS of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of SS is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, pp-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary pp-groups of rank at most 22, paralleling and building on recent results on this problem for the Olson constant

    Inverse zero-sum problems and algebraic invariants

    Full text link
    In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via the resolution of a linear integer program, a result of W. Gao and A. Geroldinger concerning the minimal number of elements with maximal order in a long zero-sumfree sequence of a finite Abelian group of rank two.Comment: 17 pages, to appear in Acta Arithmetic

    A nullstellensatz for sequences over F_p

    Get PDF
    Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.Comment: 23 page

    The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup

    Get PDF
    Let GG be a finite group written multiplicatively. By a sequence over GG, we mean a finite sequence of terms from GG which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of GG. The small Davenport constant d(G)\mathsf d (G) is the maximal integer \ell such that there is a sequence over GG of length \ell which has no nontrivial, product-one subsequence. The large Davenport constant D(G)\mathsf D (G) is the maximal length of a minimal product-one sequence---this is a product-one sequence which cannot be factored into two nontrivial, product-one subsequences. It is easily observed that d(G)+1D(G)\mathsf d(G)+1\leq \mathsf D(G), and if GG is abelian, then equality holds. However, for non-abelian groups, these constants can differ significantly. Now suppose GG has a cyclic, index 2 subgroup. Then an old result of Olson and White (dating back to 1977) implies that d(G)=12G\mathsf d(G)=\frac12|G| if GG is non-cyclic, and d(G)=G1\mathsf d(G)=|G|-1 if GG is cyclic. In this paper, we determine the large Davenport constant of such groups, showing that D(G)=d(G)+G\mathsf D(G)=\mathsf d(G)+|G'|, where G=[G,G]GG'=[G,G]\leq G is the commutator subgroup of GG

    Arithmetic-Progression-Weighted Subsequence Sums

    Full text link
    Let GG be an abelian group, let SS be a sequence of terms s1,s2,...,snGs_1,s_2,...,s_{n}\in G not all contained in a coset of a proper subgroup of GG, and let WW be a sequence of nn consecutive integers. Let WS={w1s1+...+wnsn:  wiatermofW,wiwjforij},W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\}, which is a particular kind of weighted restricted sumset. We show that WSmin{G1,n}|W\odot S|\geq \min\{|G|-1,\,n\}, that WS=GW\odot S=G if nG+1n\geq |G|+1, and also characterize all sequences SS of length G|G| with WSGW\odot S\neq G. This result then allows us to characterize when a linear equation a1x1+...+arxrαmodn,a_1x_1+...+a_rx_r\equiv \alpha\mod n, where α,a1,...,arZ\alpha,a_1,..., a_r\in \Z are given, has a solution (x1,...,xr)Zr(x_1,...,x_r)\in \Z^r modulo nn with all xix_i distinct modulo nn. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group GCn1Cn2G\cong C_{n_1}\oplus C_{n_2} (where n1n2n_1\mid n_2 and n23n_2\geq 3) having kk distinct terms, for any k[3,min{n1+1,exp(G)}]k\in [3,\min\{n_1+1,\,\exp(G)\}]. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence
    corecore