{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:42:32Z","timestamp":1750308152702,"version":"3.41.0"},"reference-count":7,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2005,9,1]],"date-time":"2005-09-01T00:00:00Z","timestamp":1125532800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["SIGSAM Bull."],"published-print":{"date-parts":[[2005,9]]},"abstract":"<jats:p>A multivariate hyperexponential function is a function whose\n\"logarithmic derivatives\" are rational. Examples of\nhyperexponential functions include rational functions, exponential\nfunctions, and hypergeometric terms. Hyperexponential functions\nplay an important role in the handling of analytic and\ncombinatorial objects. We present a few algorithms applicable to\nthe manipulation of hyperexponential functions in an uniform\nway.<\/jats:p>\n          <jats:p>Let &lt;i&gt;F&lt;\/i&gt; be a field of characteristic zero, on\nwhich derivation operators\n&amp;delta;&lt;inf&gt;1&lt;\/inf&gt;,...,&amp;delta;&lt;inf&gt;&amp;ell;&lt;\/inf&gt;\nand difference operators (automorphisms)\n&amp;sigma;&lt;inf&gt;&amp;ell;+1&lt;\/inf&gt;,...,\n&amp;sigma;&lt;inf&gt;m&lt;\/inf&gt; act. Let &lt;i&gt;E&lt;\/i&gt;\nbe an &lt;i&gt;F&lt;\/i&gt;-algebra. Assume that the\n&amp;delta;&lt;inf&gt;&lt;i&gt;i&lt;\/i&gt;&lt;\/inf&gt; for 1\n&amp;le; &lt;i&gt;i&lt;\/i&gt; &amp;le; &amp;ell; and\n&amp;sigma;&lt;inf&gt;&lt;i&gt;j&lt;\/i&gt;&lt;\/inf&gt; for\n&amp;ell; + 1 &amp;le; &lt;i&gt;m&lt;\/i&gt; can be extended to\n&lt;i&gt;E&lt;\/i&gt; as derivation and difference operators.\nMoreover, these operators commute with each other on\n&lt;i&gt;E.&lt;\/i&gt; A hyperexponential element of\n&lt;i&gt;E&lt;\/i&gt; over &lt;i&gt;F&lt;\/i&gt; is defined to be a\nnonzero element &lt;i&gt;h&lt;\/i&gt; &amp;isin;\n&lt;i&gt;E&lt;\/i&gt; such that<\/jats:p>\n          <jats:p>&amp;delta;&lt;inf&gt;1&lt;\/inf&gt;(&lt;i&gt;h&lt;\/i&gt;) =\n&lt;i&gt;r&lt;\/i&gt;&lt;inf&gt;1&lt;\/inf&gt;&lt;i&gt;h&lt;\/i&gt;,\n...,&amp;delta;&lt;inf&gt;&amp;ell;&lt;\/inf&gt;(&lt;i&gt;h&lt;\/i&gt;)\n=\n&lt;i&gt;r&lt;\/i&gt;&lt;inf&gt;&amp;ell;&lt;\/inf&gt;&lt;i&gt;h&lt;\/i&gt;,\n&amp;sigma;&lt;inf&gt;&amp;ell;+1&lt;\/inf&gt;(&lt;i&gt;h&lt;\/i&gt;)\n=\n&lt;i&gt;r&lt;\/i&gt;&lt;inf&gt;&amp;ell;+1&lt;\/inf&gt;&lt;i&gt;h&lt;\/i&gt;,...,&amp;sigma;&lt;inf&gt;&lt;i&gt;m&lt;\/i&gt;&lt;\/inf&gt;(&lt;i&gt;h&lt;\/i&gt;)\n=\n&lt;i&gt;r&lt;\/i&gt;&lt;inf&gt;&lt;i&gt;m&lt;\/i&gt;&lt;\/inf&gt;&lt;i&gt;h&lt;\/i&gt;<\/jats:p>\n          <jats:p>for some &lt;i&gt;r&lt;\/i&gt;&lt;inf&gt;1&lt;\/inf&gt;,...,\n&lt;i&gt;r&lt;inf&gt;m&lt;\/inf&gt;&lt;\/i&gt; &amp;isin;\n&lt;i&gt;F&lt;\/i&gt;. These rational functions are called\n(rational) certificates for &lt;i&gt;h.&lt;\/i&gt;<\/jats:p>","DOI":"10.1145\/1113439.1113446","type":"journal-article","created":{"date-parts":[[2007,1,17]],"date-time":"2007-01-17T18:32:02Z","timestamp":1169058722000},"page":"84-85","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Computation with hyperexponential functions"],"prefix":"10.1145","volume":"39","author":[{"given":"Ziming","family":"Li","sequence":"first","affiliation":[{"name":"Chinese Academy of Sciences, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dabin","family":"Zheng","sequence":"additional","affiliation":[{"name":"Chinese Academy of Sciences, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2005,9]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.2002.0529"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/860854.860861"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/1005285.1005313"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.1998.0251"},{"volume-title":"Academic Press","year":"1973","author":"Kolchin E. R.","key":"e_1_2_1_5_1"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/1005285.1005317"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(03)00090-7"}],"container-title":["ACM SIGSAM Bulletin"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1113439.1113446","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1113439.1113446","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T16:18:47Z","timestamp":1750263527000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1113439.1113446"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2005,9]]}},"alternative-id":["10.1145\/1113439.1113446"],"URL":"https:\/\/doi.org\/10.1145\/1113439.1113446","relation":{},"ISSN":["0163-5824"],"issn-type":[{"type":"print","value":"0163-5824"}],"subject":[],"published":{"date-parts":[[2005,9]]},"assertion":[{"value":"2005-09-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}