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In this article, the properties of being equational noetherian, q ω and u ω -compactness, and equational Artinian are studied from the perspective of the Zariski topology. The equational conditions on the relative free algebras of arbitrary varieties are also investigated and their relations to some logic and model theory notions are obtained. Some applications for the case of the universal algebraic geometry over groups are also introduced.

This article is devoted to the number of non-negative solutions of the linear Diophantine equationwhere a1, . . . , an, and d are positive integers. We obtain a relation between the number of solutions of this equation and characters of the symmetric group, using relative symmetric polynomials. As an application, we give a necessary and sufficient condition for the space of the relative symmetric polynomials to be non-zero.