{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-goncharov-depth-conjecture-and-a","title":"On the Goncharov depth conjecture and a formula for volumes of orthoschemes","arxiv_id":"2012.05599","date":"2020-12-10","proceeding":null,"authors":["Daniil Rudenko"],"abstract":"We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Our second result is a formula for volume of hyperbolic orthoschemes, generalizing the formula of Lobachevsky in dimension $3$ to an arbitrary dimension. We show a surprising relation between two results, which comes from the fact that hyperbolic orthoschemes are parametrized by configurations of points on $\\mathbb{P}^1.$ In particular, we derive both formulas from their common generalization.","url_abs":"https://arxiv.org/abs/2012.05599v2","url_pdf":"https://arxiv.org/pdf/2012.05599v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-goncharov-depth-conjecture-and-a","repo_url":"https://github.com/amatveiakin/polykit","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"on-the-goncharov-depth-conjecture-and-a","repo_url":"https://github.com/amatveyakin/polykit","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}