{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T10:16:00Z","timestamp":1760523360529,"version":"3.41.0"},"reference-count":8,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2015,11,24]],"date-time":"2015-11-24T00:00:00Z","timestamp":1448323200000},"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":["ACM Commun. Comput. Algebra"],"published-print":{"date-parts":[[2015,11,24]]},"abstract":"\n The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[\n x<\/jats:italic>\n 1<\/jats:sub>\n , ...,\n \n x\n k<\/jats:sub>\n <\/jats:italic>\n ] have been extensively studied since the famous paper of Hilbert:\n Ueber die Theorie der algebraischen Formen<\/jats:italic>\n (\n [Hilbert, 1890])<\/jats:italic>\n . In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi-polynomial\n P<\/jats:italic>\n of degree\n n<\/jats:italic>\n splits into a polynomial\n S<\/jats:italic>\n of degree\n n<\/jats:italic>\n and a lower degree quasi-polynomial\n T<\/jats:italic>\n . We have completely determined the degree of\n T<\/jats:italic>\n and the first few coefficients of\n P<\/jats:italic>\n . Moreover, the quasi-polynomial\n T<\/jats:italic>\n has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[\n x<\/jats:italic>\n 1<\/jats:sub>\n , ...,\n \n x\n k<\/jats:sub>\n <\/jats:italic>\n ]\/\n I<\/jats:italic>\n .\n <\/jats:p>","DOI":"10.1145\/2850449.2850461","type":"journal-article","created":{"date-parts":[[2015,11,30]],"date-time":"2015-11-30T19:03:44Z","timestamp":1448910224000},"page":"101-104","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["On the Hilbert quasi-polynomials for non-standard graded rings"],"prefix":"10.1145","volume":"49","author":[{"given":"Massimo","family":"Caboara","sequence":"first","affiliation":[{"name":"Universit\u00e0 di Pisa, Pisa, Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carla","family":"Mascia","sequence":"additional","affiliation":[{"name":"Universit\u00e0 di Pisa, Pisa, Pisa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2015,11,24]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(96)00035-7"},{"key":"e_1_2_1_2_1","first-page":"1305","volume":"5","author":"Bruns W.","year":"2007","journal-title":"Number"},{"key":"e_1_2_1_3_1","doi-asserted-by":"crossref","unstructured":"M. Caboara C. Mascia On the Hilbert quasi-polynomials of non-standard graded rings in preparation 2015. M. Caboara C. Mascia On the Hilbert quasi-polynomials of non-standard graded rings in preparation 2015.","DOI":"10.1145\/2850449.2850461"},{"key":"e_1_2_1_4_1","unstructured":"L. Cirillo Propriet\u00e0 e calcolo dei polinomi di Hilbert in graduazione non standard Bachelor Thesis Universit\u00e0 di Pisa 2014. L. Cirillo Propriet\u00e0 e calcolo dei polinomi di Hilbert in graduazione non standard Bachelor Thesis Universit\u00e0 di Pisa 2014."},{"key":"e_1_2_1_5_1","unstructured":"C. Mascia On the Hilbert quasi-polynomials of non-standard graded rings Master Thesis Universit\u00e0 di Pisa 2014. C. Mascia On the Hilbert quasi-polynomials of non-standard graded rings Master Thesis Universit\u00e0 di Pisa 2014."},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/1837934.1837961"},{"key":"e_1_2_1_7_1","unstructured":"Decker W.; Greuel G.-M.; Pfister G.; Sch\u00f6nemann H.: Singular 4-0-2 --- A computer algebra system for polynomial computations. http:\/\/www.singular.uni-kl.de (2015). Decker W.; Greuel G.-M.; Pfister G.; Sch\u00f6nemann H.: Singular 4-0-2 --- A computer algebra system for polynomial computations. http:\/\/www.singular.uni-kl.de (2015)."},{"key":"e_1_2_1_8_1","doi-asserted-by":"crossref","unstructured":"W. Vasconcelos Computational Methods in Commutative Algebra and Algebraic Geometry Springer 1997. W. Vasconcelos Computational Methods in Commutative Algebra and Algebraic Geometry Springer 1997.","DOI":"10.1007\/978-3-642-58951-5"}],"container-title":["ACM Communications in Computer Algebra"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2850449.2850461","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2850449.2850461","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T05:43:27Z","timestamp":1750225407000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2850449.2850461"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,24]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,11,24]]}},"alternative-id":["10.1145\/2850449.2850461"],"URL":"https:\/\/doi.org\/10.1145\/2850449.2850461","relation":{},"ISSN":["1932-2240"],"issn-type":[{"type":"print","value":"1932-2240"}],"subject":[],"published":{"date-parts":[[2015,11,24]]},"assertion":[{"value":"2015-11-24","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}