Powers of unary ideals rarely have minimum Taylor resolution, even in the squareless case. In this paper, we introduce a smaller resolution for each power of any square free monomial ideal that depends only on the number of ideal generators. More precisely, for every pair of fixed integers $r$ and $q$, we have a simplicial complex that supports free resolution of any square-free unary ideal to the $r$ power using the $q$ generator. build the body The resulting resolution is significantly smaller than the Taylor resolution and minimal in special cases. These resolutions can be further reduced if we take into account the relationship regarding the origin of the fixed ideal. We also introduce a class of ideals called “extremum ideals” and show that the betty powers of all square-free unary ideals are bounded by the powers of the polar ideals. Our results lead to an upper bound on the Betty number power of any square free unary ideal that significantly improves the binomial bound provided by the Taylor resolution.
