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We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These homotopies take advantage of Anderson\u2019s flat degeneration to a toric variety. When Anderson\u2019s degeneration embeds into projective space, our algorithm is a special case of a general toric two-step homotopy algorithm. When Anderson\u2019s degeneration is embedded in a weighted projective space, we explain how to lift to a projective space and construct an appropriate modification of the toric homotopy. Our algorithms are illustrated on several examples using Macaulay2.<\/p>","DOI":"10.1090\/mcom\/3689","type":"journal-article","created":{"date-parts":[[2021,8,4]],"date-time":"2021-08-04T13:14:01Z","timestamp":1628082841000},"page":"2333-2353","source":"Crossref","is-referenced-by-count":6,"title":["Numerical homotopies from Khovanskii bases"],"prefix":"10.1090","volume":"92","author":[{"given":"M.","family":"Burr","sequence":"first","affiliation":[]},{"given":"F.","family":"Sottile","sequence":"additional","affiliation":[]},{"given":"E.","family":"Walker","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,3,24]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1007\/s00029-005-0396-8","article-title":"Toric degenerations of spherical varieties","volume":"10","author":"Alexeev, Valery","year":"2004","journal-title":"Selecta Math. 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