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This bound was thought to be very vast because it is exponential on $d$. Indeed, all the examples we have found in the literature verify that $\\operatorname{rank}(M_A (\\beta))<2 \\vol (A)$. We construct here, in a very elementary way, some families of matrices $A_{(d)}\\in \\ZZ^{d \\times n}$ and paramete"},"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":"1201.5090","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-01-24T18:56:38Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"44f2813bf0243688deeb3e827e1ce96541857dbe9b7d4abc5be881a32841a160","abstract_canon_sha256":"8b5ef9aab8296bdabfa1e69cf4c24eefd5711eff7c98c9d087b739f641be0105"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:55.054527Z","signature_b64":"IQPkGWgZrJzplwSE2r5OS0+DLouN/V0pExyjjZAVDb1PHqHL3WQA0E5TzelFDVXXpj/Q66iqGyquXr4lFyxICw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e03e750cf89905137a924243491ec1cbdf2b1d431cb190d2672dcb142c0291c","last_reissued_at":"2026-05-18T01:10:55.053788Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:55.053788Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exponential growth of rank jumps for A-hypergeometric systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Mar\\'ia-Cruz Fern\\'andez-Fern\\'andez","submitted_at":"2012-01-24T18:56:38Z","abstract_excerpt":"The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of a $A$--hypergeometric system $M_A (\\beta)$ is known to be bounded above by $ 2^{2d}\\operatorname{vol}(A)$, where $d$ is the rank of the matrix $A$ and $\\vol (A)$ is its normalized volume. 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