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The aim of this paper is:  \\begin{enumerate}\\item Give an effective  pseudo-polynomial algorithm on $a\\_1$,  which computes The Ap{\\'e}ry set and the Frobenius number of $S$. As a consequence it also solves in pseudo-polynomial time the  integer knapsack problem : given a natural integer b, b belongs to $S$?\\item The \\gbb of $I$ for the reverse lexicographic order to $x\\_n,\\ldots ,x\\_1$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1510.01973","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-10-07T14:53:33Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"afc415371d14bf456b2faed2faaa71ba130aa9a67e94c92354665ae3ee666469","abstract_canon_sha256":"ccfe6384a70ba53e005d265d71d318f499500f5699f6b65d4f968a98e1976d60"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:06.425200Z","signature_b64":"Sqm+cCMbJQgcYivHJoEw2aaMtocWKz0mX5r3dxATCwflkvyqF/Ss52eSAaevr3fyGRaI81FwseSdPwelNsVxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c6652f2c6f722d73d30fb502925501fb5e7abb4c561bbe7f0e56a27cca55f3e","last_reissued_at":"2026-05-18T01:24:06.424511Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:06.424511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gr{\\\"o}bner basis. a \"pseudo-polynomial\" algorithm for computing the Frobenius number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Dung Nguyen Thi, Marcel Morales","submitted_at":"2015-10-07T14:53:33Z","abstract_excerpt":"Let consider $n$ natural numbers    $a\\_1  ,\\ldots ,  a\\_{n}  $. 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