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We expect that the inequality $|G(I^2)|>|G(I)|$ should hold and that $|G(I^i)|$, $i\\ge 2$, grows further whenever $|G(I)|\\ge 2$. In this paper we will disprove this expectation and show that for any $n$ and $d$ there is an $\\mathfrak m$-primary monomial ideal $I\\subset \\mathbb K[x_1,\\ldots,x_n]$ such that $|G(I)|>|G(I^i)|$ for"},"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":"1908.10702","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-08-28T13:05:05Z","cross_cats_sorted":[],"title_canon_sha256":"9517eb4987d2f8fae7188243fff8c93be0bc6b6f16756816b3392a7e35290003","abstract_canon_sha256":"b5903781286c33a2597e8a10016a76be2dcc30a99d86185945cec31b1957ab7f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:00:35.399199Z","signature_b64":"b7LyiFKmOEljaCj45s8hiDCPuYVQcBTxxPy6tWgtQb/OxMWeyax9HCwKMh4kkcBVHdra9w4is+OU3R0/B9vhCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61fdd53841d5682757c9df9b1a753037e31c404a60fd27efc85fa7446c2a8acd","last_reissued_at":"2026-07-05T00:00:35.398794Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:00:35.398794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Monomial ideals with arbitrarily high tiny powers in any number of variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Oleksandra Gasanova","submitted_at":"2019-08-28T13:05:05Z","abstract_excerpt":"Powers of (monomial) ideals is a subject that still calls attraction in various ways. 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