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We illustrate this by examples coming from actual operator statements and discuss relevant algorithmic methods for finding such polynomials based on noncommutative Gr\u00f6bner bases. In particular, we present algorithms for computing the intersection of a two-sided ideal with a one-sided ideal as well as for computing homogeneous polynomials in two-sided ideals and monomials in one-sided ideals. All methods presented in this work are implemented in the <jats:sc>Mathematica<\/jats:sc> package .<\/jats:p>","DOI":"10.1007\/s11786-022-00536-5","type":"journal-article","created":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T03:23:01Z","timestamp":1663730581000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Computing Elements of Certain Form in Ideals to Prove Properties of Operators"],"prefix":"10.1007","volume":"16","author":[{"given":"Clemens","family":"Hofstadler","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Clemens G.","family":"Raab","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Georg","family":"Regensburger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,21]]},"reference":[{"key":"536_CR1","doi-asserted-by":"publisher","first-page":"1194","DOI":"10.1016\/j.laa.2010.05.001","volume":"433","author":"ML Arias","year":"2010","unstructured":"Arias, M.L., Gonzalez, M.C.: Positive solutions to operator equations $$AXB = C$$. 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