Related papers: Classifications of $\ell$-Zero-Sumfree Sets

In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition…

Let $\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of $A$ ? (2) When can…

Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not…

Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $\omega(N)\to\infty$ as…

For any $\ell > 0$, we present an algorithm which takes as input a semi-algebraic set, $S$, defined by $P_1 \leq 0,...,P_s \leq 0$, where each $P_i \in \R[X_1,...,X_k]$ has degree $\leq 2,$ and computes the top $\ell$ Betti numbers of $S$,…

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…

A set A of integers is said to be sum-free if there are no solutions to the equation x + y = z with x,y and z all in A. Answering a question of Cameron and Erdos, we show that the number of sum-free subsets of {1,...,N} is O(2^(N/2)).

A vector $(v_{1}, v_{2}, \cdots, v_{d})$ in $\mathbb{Z}_n^{d}$ is said to be a zero-sum-free $d$-tuple if there is no non-empty subset of its components whose sum is zero in $\mathbb{Z}_n$. We denote the cardinality of this collection by…

Let $\Delta$ be a simplicial complex on $[n]$. The $\mathcal{N}\mathcal{F}$-complex of $\Delta$ is the simplicial complex $\delta_{\mathcal{N}\mathcal{F}}(\Delta)$ on $[n]$ for which the facet ideal of $\Delta$ is equal to the…

We are interested in ordering the elements of a subset A of the non-zero integers modulo n in such a way that all the partial sums are distinct. We conjecture that this can always be done and we prove various partial results about this…

We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, and $\zeta$. Particularly,…

Let $M$ be a commutative cancellative monoid. The set $\Delta(M)$, which consists of all positive integers which are distances between consecutive factorization lengths of elements in $M$, is a widely studied object in the theory of…

This paper is a contribution to the study of the geometry of algebras related the Weyl groupoid initiated in \cite{M22}. The Nullstellensatz gives a bijection between radical ideals of such an algebra and their zero loci, the superalgebraic…

Classification, up to isomorphism, of algebras from a non-empty subset of the variety of $n$- dimensional algebras is presented. It is shown that these algebras have only trivial automorphism and if the basic field is algebraically closed…

We introduce multi-colour partition algebras $P_{n,m}(\delta_0, ..., \delta_{m-1})$, which are generalization of both bubble algebras and partition algebras, then define the bubble algebra $T_{n,m}(\delta_0, ..., \delta_{m-1})$ as a…

A numerical semigroup $S$ is a cofinite, additively-closed subset of $\mathbb Z_{\ge 0}$ that contains 0, and a factorization of $x \in S$ is a $k$-tuple $z = (z_1, \ldots, z_k)$ where $x = z_1a_1 + \cdots + z_ka_k$ expresses $x$ as a sum…

A finite subset $A$ of an abelian group $G$ is said to be zero-free if the identity element of $G$ cannot be written as a sum of distinct elements from $A$. In this article we study the structure of zero-free subsets of $Z/pZ$ the…

Let L be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of L. In this paper we introduce a notion…

A $k$-wise $\ell$-divisible set family is a collection $\mathcal{F}$ of subsets of ${ \{1,\ldots,n \} }$ such that any intersection of $k$ sets in $\mathcal{F}$ has cardinality divisible by $\ell$. If $k=\ell=2$, it is well-known that…

An $s$-subset of codewords of a binary code $X$ is said to be an {\em $(s,\ell)$-bad} in $X$ if the code $X$ contains a subset of other $\ell$ codewords such that the conjunction of the $\ell$ codewords is covered by the disjunctive sum of…