Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Abstract

Let ( G , + ) be an abelian group and consider a subset A ⊆ G with ∣ A ∣ = k. Given an ordering ( a 1 , … , a k ) of the elements of A, define its partial sums by s 0 = 0 and s j = ∑ i = 1 j a i for 1 ≤ j ≤ k. We consider the following conjecture of Alspach: for any cyclic group Z n and any subset A ⊆ Z n ⧹ { 0 } with s k ≠ 0, it is possible to find an ordering of the elements of A such that no two of its partial sums s i and s j are equal for 0 ≤ i < j ≤ k. We show that Alspach’s Conjecture holds for prime n when k ≥ n − 3 and when k ≤ 10. The former result is by direct construction, the latter is nonconstructive and uses the polynomial method. We also use the polynomial method to show that for prime n a sequence of length k having distinct partial sums exists in any subset of Z n ⧹ { 0 } of size at least 2 k − 8 k in all but at most a bounded number of cases.

Some new results about a conjecture by Brian Alspach

“…In Section 4, we prove the validity of Conjecture 1.2 for subsets of size 12 of cyclic groups of prime order. This result is achieved using Alon's combinatorial Nullstellensatz and the techniques developed in [17]. As a consequence, we obtain a similar extension to torsion-free abelian groups and a similar asymptotic result.…”

“…In [17], the validity of Conjecture 1.1 was proved for any cyclic group Z p , where p is a prime, whenever k = |A| ≤ 10. This result was achieved using a polynomial method based on Alon's combinatorial Nullstellensatz.…”

“…Now, consider the monomial We can use Alon's combinatorial Nullstellensatz to prove the G-ADMS conjecture. Also in this case, we use the polynomial provided by [17]. For any k ≥ 2, let…”

“…where 1 ≤ j ≤ k and c k,j ∈ Z. In[17, Table 1], the authors described the coefficients c k,j for k ≤ 10, showing that either gcd(c k,1 , . .…”

Sequences in Dihedral Groups with Distinct Partial Products

“…Perhaps more importantly, it embodies an approach that could plausibly be extended to all values of k in these groups. This generalises a method developed for cyclic groups in [18].…”