[
    {
        "id": "osp-13835",
        "type": "article-journal",
        "title": "Finite Sets with Large Algebraic Closures in Nilpotent Groups",
        "author": [
            {
                "family": "Al Raisi",
                "given": "O."
            },
            {
                "family": "Shahryari",
                "given": "M."
            }
        ],
        "URL": "https://omanscience.com/ar/articles/finite-sets-with-large-algebraic-closures-in-nilpotent-groups",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2025
                ]
            ]
        },
        "container-title": "Sultan Qaboos University Journal for Science",
        "volume": "30",
        "issue": "2",
        "page": "84-87",
        "DOI": "10.53539/2414-536x.1402",
        "publisher": "Sultan Qaboos University",
        "ISSN": "2308-3921",
        "abstract": "Algebraic sets play a central role in algebraic geometry over groups; by definition, an algebraic set is the set of common roots of a collection of equations in some non-commuting variables x1, ..., xn having coefficients in some group G. Often, one seeks solutions of equations over G in the affine n-space Gn; such systems of polynomial equations are termed Diophantine systems. In this brief note, we consider Diophantine systems over a group G that is assumed to be nilpotent of class c and to have an element of infinite order in its center; we associate with each n × n invertible matrix A having integral coefficients an algebraic set YAac and prove that (γc(G))n ⊆ YAac; here, γc(G) stands for the subgroup of G generated by all simple commutators of length c + 1. Consequently, we prove that if G is an abelian group having an element of infinite order, then there exist infinitely many subsets of the affine n-space Gn having cardinality n + 1 that are algebraically dense in Gn; in particular, this provides us with another proof of the well-known fact that the union of two algebraic sets need not be an algebraic set."
    }
]