الملخص
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.
الكلمات المفتاحية
بيانات النشر
- المعرّف الرقمي
- 10.53539/2414-536x.1402
- المجلة
- مجلة جامعة السلطان قابوس للعلوم, 30(2), 84-87
- الناشر
- جامعة السلطان قابوس
- وصول مفتوح
- وصول مفتوح ذهبي
- الترخيص
- CC BY 4.0
اقتبس هذه المقالة
APA 7
Al Raisi, O., & Shahryari, M. (2025). Finite Sets with Large Algebraic Closures in Nilpotent Groups. Sultan Qaboos University Journal for Science, 30(2), 84-87. https://doi.org/10.53539/2414-536x.1402
MLA 9
Al Raisi, O., and M. Shahryari. "Finite Sets with Large Algebraic Closures in Nilpotent Groups." Sultan Qaboos University Journal for Science, vol. 30, no. 2, 2025, pp. 84-87. https://doi.org/10.53539/2414-536x.1402.
شيكاغو (المؤلف–التاريخ)
Al Raisi, O., and M. Shahryari. 2025. "Finite Sets with Large Algebraic Closures in Nilpotent Groups." Sultan Qaboos University Journal for Science 30 (2): 84-87. https://doi.org/10.53539/2414-536x.1402.
هارفارد
Al Raisi, O. and Shahryari, M. (2025) 'Finite Sets with Large Algebraic Closures in Nilpotent Groups', Sultan Qaboos University Journal for Science, 30(2), pp. 84-87. doi:10.53539/2414-536x.1402.
فانكوفر
Al Raisi O, Shahryari M. Finite Sets with Large Algebraic Closures in Nilpotent Groups. Sultan Qaboos University Journal for Science. 2025;30(2):84-87. doi:10.53539/2414-536x.1402
IEEE
O. Al Raisi, and M. Shahryari, "Finite Sets with Large Algebraic Closures in Nilpotent Groups," Sultan Qaboos University Journal for Science, vol. 30, no. 2, pp. 84-87, 2025, doi: 10.53539/2414-536x.1402.