Some new results about a conjecture by Brian Alspach

In this paper, we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset A of Zn\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_n{\setminus } \{0\}$$\end{document} of size k such that ∑z∈Az≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{z\in A} z\not = 0$$\end{document}, it is possible to find an ordering (a1,…,ak)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_1,\ldots ,a_k)$$\end{document} of the elements of A such that the partial sums si=∑j=1iaj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_i=\sum _{j=1}^i a_j$$\end{document}, i=1,…,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,k$$\end{document}, are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size k≤11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\le 11$$\end{document} in cyclic groups of prime order. Here, we extend this result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_n$$\end{document}. We also consider a related conjecture, originally proposed by Ronald Graham: given a subset A of Zp\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_p{\setminus }\{0\}$$\end{document}, where p is a prime, there exists an ordering of the elements of A such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis, and Schmitt, based on Alon’s combinatorial Nullstellensatz, we prove the validity of this conjecture for subsets A of size 12.

In Section 2, we explain how the validity of Conjecture 1.1 for sets of size k in cyclic groups Z p , for infinitely many primes p, implies the validity for sets of size k in any torsion-free abelian group. As a consequence, in Section 3, we provide an asymptotic result for sets of size k ≤ 11 in finite cyclic groups: this has been achieved without any direct or recursive construction (that we believe can hardly be obtained) but only with some theoretical nonconstructive arguments.
Another conjecture, very close to the Alspach one, was originally proposed by R.L. Graham in [16] for cyclic groups of prime order, and by D.S. Archdeacon, J.H. Dinitz, A. Mattern, and D.R. Stinson for any finite cyclic group, see [6]. In [6], the authors proved that Conjecture 1.1 in a group Z n for sets of size at most k implies Conjecture 1.2 in the same group Z n for sets of size at most k. As remarked in [11], the G-ADMS conjecture can be extended to any finite subset of an abelian group. This implies that our results on Alspach's conjecture can also be applied to the G-ADMS conjecture. 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.

2.
Alspach's conjecture for torsion-free abelian groups. In this paper, we will say that a finite subset A of an abelian group (G, +) is nice if 0 G ∈ A and z∈A z = 0 G . Also, with an abuse of notation, we will say that Alspach's conjecture is true in G for any subset of size k if it is true for any nice subset of size k.
Given a nice subset A of an abelian group G, by Δ(A) we mean the set {x 1 − x 2 : x 1 , x 2 ∈ A, x 1 = x 2 }. This allows us to define the set Given two integers a, b with a ≤ b, the subset {a, a + 1, . . . , b} ⊂ Z will be denoted by [a, b]. Our aim is to extend the known results on Alspach's conjecture in cyclic groups of prime order to any torsion-free abelian group. Lemma 2.1. Let G 1 and G 2 be abelian groups such that Alspach's conjecture holds in G 2 for any subset of size k. Given a nice subset A of G 1 of size k, suppose there exists an homomorphism ϕ : G 1 → G 2 such that ker(ϕ)∩Υ(A) = ∅. Then Alspach's conjecture is true for the subset A.
Proof. Since no element of Υ(A) belongs to the kernel of ϕ, ϕ(A) is a nice subset of G 2 of size k. Therefore, there exists an ordering ω 2 = (x 1 , x 2 , . . . , x k ) of the elements of ϕ(A) such that s i (ω 2 ) = 0 G2 and s i (ω 2 ) = s j (ω 2 ) for all 1 ≤ i < j ≤ k. However, considering the ordering ω 1 = (z 1 , z 2 , . . . , z k ) of the elements of A, where ϕ(z i ) = x i for all i = 1, . . . , k, we obtain that the partial sums s i (ω 1 ) in G 1 are still pairwise distinct and nonzero.

Proposition 2.2.
Let k be a positive integer and suppose that, for infinitely many primes p, Alspach's conjecture holds in Z p for any subset of size k. Then Alspach's conjecture holds in Z for any subset of size k.
Proof. Consider a nice subset A of Z of size k. Let p > max z∈Υ(A) |z| be a prime such that Alspach's conjecture holds in Z p for subsets of size k. Then Υ(A) and the kernel of the canonical projection π p : Z → Z p are disjoint sets. Hence, the statement follows from Lemma 2.1.
We now consider the free abelian group Z n of rank n.  [1,n] n|z j |, we define the homomorphism ϕ : Z n → Z as follows: Because of the choice of M , the subset B and the kernel of ϕ are disjoint. Namely, suppose that there exists b = (y 1 , y 2 , . . . , y n ) ∈ B such that ϕ(b) = 0. We can assume that y s1 , . . . , y sc are all nonnegative integers and that y t1 , . . . , y t d are all negative integers. Then we can write We can look at the two sides of this equality as two expansions in base M of the same nonnegative integer since the coefficients y s1 , . . . , y sc , y t1 , . . . , y t d all belong to the set [0, M − 1]. The uniqueness of such an expansion implies that all these coefficients are zero, i.e., that b = 0. It follows from Lemma 2.1 that Alspach's conjecture holds for the subset A.
From the previous proposition, we deduce this result. 3. An asymptotic result. Given an element g of an abelian group G, we denote by o(g) the cardinality of the cyclic subgroup g generated by g. Furthermore, we set Now we are ready to prove that, if ϑ(G) is large enough, Alspach's conjecture is true for k ≤ 11. This result can be deduced from the compactness theorem of the first order logic but, here, we give a more direct proof. Now we want to prove that H is torsion-free. Let us suppose, for sake of contradiction, that there exists an element [0] = [x] ∈ H of finite order, say n. Let π j : G → G Mj be the canonical projection on G Mj . For any i such that M i > n, either π i (x) = 0 GM i or we have n · π i (x) = 0 GM i . However, since n · [x] = [0] in H and due to the definition of ≈, we should have n · π i (x) = 0 GM i for i large enough. It follows that π i (x) is eventually zero but this is a contradiction since [x] is nonzero. Therefore, H is torsion-free. Now we consider the following subset A of H: Clearly

Implications on other conjectures.
We consider here two conjectures related to the Alspach one. These conjectures have recently been studied mainly in relation to (relative) Heffter arrays and their application for constructing cyclic cycle decompositions of (multipartite) complete graphs, see [4,5,9,[12][13][14].

The G-ADMS conjecture.
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. A 1 , . . . , a k ∈ A k so that f (a 1 , . . . , a k ) = 0.

is a nonnegative integer, and suppose the coefficient of
In order to apply this theorem for proving Alspach's conjecture for subsets of Z p of size k, J. Hicks, M.A. Ollis, and J.R. Schmitt constructed a suitable homogeneous polynomial F k of degree k(k−1)−1, identifying a monomial with nonzero coefficient such that the degree of each of its terms x i is less than |A| = k (clearly, we may assume p > k). The existence of values a 1 , . . . , a k ∈ A such that F k (a 1 , . . . , a k ) = 0, given by Theorem 4 .1, implies that ω = (a 1 , . . . , a k ) is an ordering of the elements of A satisfying the requirements of Conjecture 1.1.
We recall here the polynomials given in [17]. For every k ≥ 2, let and Observe that we can also define F k recursively. To this purpose, define G ∈ Z[x 1 , x 2 , . . . , x +1 ], where 2 ≤ ≤ k − 1, as follows: Then F ∈ Z[x 1 , x 2 , . . . , x ] can be defined as Now, consider the monomial In [17, Table 1], the authors described the coefficients c k,j for k ≤ 10, showing that either gcd(c k,1 , . . . , c k,k ) equals 1 or its prime factors are all less than k. This means that, for any prime p > k, there exists a coefficient c k,j which is nonzero modulo p, proving the validity of Conjecture 1.1 for subsets of Z p of size k.
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 Note that f k+1 ∈ Z[x 1 , . . . , x k+1 ] is a homogeneous polynomial of degree k 2 . By Theorem 4.1, to prove the validity of Conjecture 1.2 for subsets of Z p of size k + 1, it suffices to find a monomial of f k+1 with nonzero coefficient such that the degree of each of its terms x i is less than k + 1. Instead of working directly with the polynomial f k+1 , we show how to use the polynomial F k also for proving the validity of the G-ADMS conjecture via Alon's combinatorial Nullstellensatz. The main advantage is computational: instead of working with a polynomial of degree k 2 in k + 1 indeterminates, we work with a polynomial of degree k 2 − k − 1 in k indeterminates. Hence, consider the monomial In order to apply Theorem 4.1, we would like to determine the values of the coefficients d k+1,j . To this purpose, take the monomial e k,j = e k,j · x k 1 · · · x k j−1 · x k j+1 · · · x k k of g k (x 1 , . . . , x k ), where j ∈ [1, k] and e k,j ∈ Z. If j ≥ 2, then x k 1 divides d k+1,j and so, by (4.2), the value of (−1) k · d k+1,j coincides with the coefficient of Vol. 115 (2020) Some new results about a conjecture by Brian Alspach 485 x k 2 · · · x k j−1 · x k j+1 · · · x k k+1 in g k (x 2 , . . . , x k+1 ). In other words, we obtain that d k+1,j+1 = (−1) k e k,j for all j ∈ [1, k].
Hence, the problem of computing the values of the coefficients d k+1,j+1 is equivalent to the problem of computing the coefficients e k,j of g k = (x 1 + · · · + x k ) · F k , see (4.1). So, for any i, j ∈ [1, k] such that i = j, take the monomial For instance, using the routines for Magma that we give in [10] we determine the values of a (6) i,j , see Table 1. More generally, for k ∈ [3,10], we obtain that e k,j = (−1) k−1 2 c k,j for all j ∈ [1, k].
It would be very interesting to prove this equality for every value of k.

Proposition 4.2.
The G-ADMS conjecture holds for subsets of size k ≤ 12 of cyclic groups of prime order.
One can easily adjust to the G-ADMS conjecture the arguments of Sections 2 and 3 . Clearly, some small modifications are required. In particular, given an abelian group (G, +), a finite subset A of G is nice for the G-ADMS conjecture if 0 G ∈ A; also, given a nice subset A of G, let Υ(A) = A ∪ Δ(A).
Hence, from Proposition 4.2, we can deduce the following results.  Table 1. Values of the coefficients a (6) i,j