{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T08:11:30Z","timestamp":1768896690094,"version":"3.49.0"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2020,2,28]],"date-time":"2020-02-28T00:00:00Z","timestamp":1582848000000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,2,28]],"date-time":"2020-02-28T00:00:00Z","timestamp":1582848000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,2,28]],"date-time":"2020-02-28T00:00:00Z","timestamp":1582848000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,4,1]]},"abstract":"<jats:p>An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function $\\varphi$ on the reals, and for any positive continuous function $\\epsilon(t)$, it has a $\\mathcal{C}^\\infty$ solution with $| y(t) - \\varphi(t) | &amp;lt; \\epsilon(t)$ for all $t$. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, Rubel's DAE \\emph{never} has a unique solution, even with a finite number of conditions of the form $y^{(k_i)}(a_i)=b_i$.   The question whether one can require the solution that approximates $\\varphi$ to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a \\textbf{fixed} polynomial ODE such that for any $\\varphi$ and $\\epsilon(t)$ there exists some initial condition that yields a solution that is $\\epsilon$-close to $\\varphi$ at all times.   In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.<\/jats:p>","DOI":"10.23638\/lmcs-16(1:28)2020","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:37:35Z","timestamp":1743701855000},"source":"Crossref","is-referenced-by-count":3,"title":["A Universal Ordinary Differential Equation"],"prefix":"10.23638","volume":"Volume 16, Issue 1","author":[{"given":"Olivier","family":"Bournez","sequence":"first","affiliation":[]},{"given":"Amaury","family":"Pouly","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2020,2,28]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1702.08328v6","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1702.08328v6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:37:35Z","timestamp":1743701855000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/4437"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,28]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-16(1:28)2020","relation":{"has-preprint":[{"id-type":"arxiv","id":"1702.08328v5","asserted-by":"subject"},{"id-type":"arxiv","id":"1702.08328v4","asserted-by":"subject"},{"id-type":"arxiv","id":"1702.08328v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1702.08328","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1702.08328","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,28]]},"article-number":"4437"}}