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Let $I\\subset \\si$ be a square free monomial ideal and $J\\subset\\sis$ a sum of scroll ideals with some extra conditions, we define the binomial extension of $I$ as $\\B=I+J\\subset \\sis$. We set $p_2(\\B)$ the minimal $i\\in\\N$ such that there exists $j>2$ such that $\\beta_{i,i+j}(\\B)\\neq 0$. In the case where J=0, Fr\\\"oberg characterized combinatorally the case $p_2(I)=\\infty$; later Eisenbud et al. solved the case $p_2(I)<\\infty$. We obtain a similar result as Fr\\\"oberg for the binomial"},"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":"1211.5364","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-11-22T20:06:34Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"5c0a3c0b6708046ccb22209c9291c57841bbbdccc1ea90971d5f970494899233","abstract_canon_sha256":"71d7281335fea1c82f4ad8ed88ff3789d9a5427afd86b95dcd4eefe95b484119"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:27.807572Z","signature_b64":"QttKXzXdjkOkWCQUTVGQzcKgTxgBIS7rnmpa/ZKE2dmPYTcg4vRiVQQG0AR25h84sDNEV2uD0ay2uc8pxZkVDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c0cc0466054ae8ae6db8038fc95e78fa5c6af5a3631d61c60ce070014e9b9d8","last_reissued_at":"2026-05-18T03:39:27.806792Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:27.806792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The ${\\rm N}_{2,p}$-property of binomial extensions of simplicial complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Hernan de Alba Casillas, Marcel Morales","submitted_at":"2012-11-22T20:06:34Z","abstract_excerpt":"M. Morales introduced a family of binomial ideals that are binomial extensions of square free monomial ideals. Let $I\\subset \\si$ be a square free monomial ideal and $J\\subset\\sis$ a sum of scroll ideals with some extra conditions, we define the binomial extension of $I$ as $\\B=I+J\\subset \\sis$. We set $p_2(\\B)$ the minimal $i\\in\\N$ such that there exists $j>2$ such that $\\beta_{i,i+j}(\\B)\\neq 0$. In the case where J=0, Fr\\\"oberg characterized combinatorally the case $p_2(I)=\\infty$; later Eisenbud et al. solved the case $p_2(I)<\\infty$. 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