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Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the standard monomials of generic ideals, we show how a Gr\\\"obner basis for the ideal $(f_1, \\dots, f_i)$ can be obtained from that of $(f_1, \\dots, f_{i-1})$. If $deg f_i = d_i$, we are able to give a complete description of the initial ideal of $I$ in the case where $d_i \\geq \\left(\\sum_{j=1}^{i-1}d_j\\right) - i -1$. It was conjectured by Moreno-Soc\\'ias that"},"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":"1711.05309","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-11-14T20:55:21Z","cross_cats_sorted":[],"title_canon_sha256":"cf633b9bc91e8ca9e65e5b321121a862505b067ef14a8bc3138cf83849ba15dd","abstract_canon_sha256":"9f632f681eb1c784cd4a21692b0bae7bd01d9a902b4a135e7025c1fec5307acf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:29.435556Z","signature_b64":"PqaINW6qtKgdDPromn8HZ6V9qZjOQRrx+HFZWS7ynNI7amGkLjehpPmipFHjFMwG05V3YHbqXjgCWVlCrKzpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25e20cdfc41d0a92642ad8432e357686ececc69d75cbbc5ef34cfbcc41099385","last_reissued_at":"2026-05-18T00:28:29.434921Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:29.434921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gr\\\"obner Bases of Generic Ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Juliane Capaverde, Shuhong Gao","submitted_at":"2017-11-14T20:55:21Z","abstract_excerpt":"Let $I = ( f_1, \\dots, f_n )$ be a homogeneous ideal in the polynomial ring $K[x_1, \\dots,x_n]$ over a field $K$ generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the standard monomials of generic ideals, we show how a Gr\\\"obner basis for the ideal $(f_1, \\dots, f_i)$ can be obtained from that of $(f_1, \\dots, f_{i-1})$. If $deg f_i = d_i$, we are able to give a complete description of the initial ideal of $I$ in the case where $d_i \\geq \\left(\\sum_{j=1}^{i-1}d_j\\right) - i -1$. 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