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Its homogeneous degree-$k$ part $c_k(n,d)$ is the $k$-th Chern class of $\\mathrm{Sym}^d(\\mathbb{C}^n)$. These Chern classes, together with their coefficients in various symmetric function bases, play a central role in enumerative geometry. Despite their simple definition, general closed formulas for their coefficients are subtle, and many structural properties of these classes have remained poorly understood.\n  In this paper we prove several conjectures concerning their structure, establish explicit formulas, and study log-concavity properties for both the Chern classes and thei","title":"Positivity in classical enumerative geometry: a case study in synchronized AI-assisted mathematics","url":"https://arxiv.org/abs/2605.25271","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.25271v1 Announce Type: cross \nAbstract: We study the symmetric polynomial $\\prod_{\\alpha\\in A_{n,d}}\\bigl(1+\\alpha_1 x_1+\\cdots+\\alpha_n x_n\\bigr)$ where $A_{n,d}:=\\{\\alpha\\in\\mathbb{Z}_{\\ge 0}^n:|\\alpha|=d\\}$, which is the total Chern class of $\\mathrm{Sym}^d(\\mathbb{C}^n)$, viewed as a torus representation whose Chern roots are the weights $\\alpha_1 x_1+\\cdots+\\alpha_n x_n$ for $\\alpha\\in A_{n,d}$. Its homogeneous degree-$k$ part $c_k(n,d)$ is the $k$-th Chern class of $\\mathrm{Sym}^d(\\mathbb{C}^n)$. These Chern classes, together with their coefficients in various symmetric function bases, play a central role in enumerative geometry. Despite their simple definition, general closed formulas for their coefficients are subtle, and many structural properties of these classes have remained poorly understood.\n  In this paper we prove several conjectures concerning their structure, establish explicit formulas, and study log-concavity properties for both the Chern classes and thei","title":"Positivity in classical enumerative geometry: a case study in synchronized AI-assisted mathematics","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-26T04:43:39Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.25271"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:188d5dc92540ecb45e255e6b52264bf38dfcc8a7ab16cc6c04a38360c99eff235826b6ac0afffcdf36fc370e43615e449dbca900fd8fa04dc5d53bbd3b523a01","signer":"crovia.substrate","subject":{"observed_at":"2026-05-26T04:43:39Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.25271"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"88bf314c155d82fd521541463d2cebc56a28cf79ccba13c157f9ac80b03bcab2","leaf_index":151666,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"cd9c816b4f054ebff10615988bb4fcf54b5ce04379c9aa798b79e62f0545f630","side":"right"},{"sibling":"98115a6a11a5dbf8d1767ecca5637e2bd36901c29c6df20f662ab1e542d81e85","side":"left"},{"sibling":"8ac07e6091b61e79029838d6919f46a80c5dd3420fdcfbcd10e08d14303505ba","side":"right"},{"sibling":"44fd28907c9ae1a5879b507c6032db46a6a45547e598806329b8480cc0b9b8f7","side":"right"},{"sibling":"3aae631a10a06a5fa7ebb7d3eed91b6efa80a684399bafdfa85c25f80cc1c983","side":"left"},{"sibling":"3e03b04fd8a45f07b370f8db25084b49432110efb495c06b9ad0d6de39b49bc1","side":"left"},{"sibling":"c91fda9088f721fc82dd878961753ba116e1e68cc711da4d9b7e39390ab3fb5b","side":"left"},{"sibling":"fd791f604d83aab5bc392a1be85abc0810167043e05b18be46d20ba5123de498","side":"right"},{"sibling":"aff657821100efc777fe98abc63d25a13c2d813d2a00f85a278e295ee3a166b3","side":"right"},{"sibling":"879666fab72e779ab55d0564eaabd64b00534fd6bba7f18412c7f31f61ffd09f","side":"right"},{"sibling":"f40ccedd90c323817e961adc0a2e2db82b8aabe192b6c9d5a373ff988987b207","side":"right"},{"sibling":"b85ea61ae405eed84392a7b6b1eee5536f5a38d6b04070638e23ec7b71e3443a","side":"right"},{"sibling":"d415e6939aee710631f5062799379b547d2c3e3d9a68f263bbb5a693285ab2ca","side":"left"},{"sibling":"e5893793e3591ed7f5e58ca94ffcfba46bb30f69fb1c25d5ba8ef49eb99f9126","side":"right"},{"sibling":"35ca36cee447f0ef7064a25d55f59357c66901e31427729c4c1d8b14aa8adb6c","side":"left"},{"sibling":"e3eecaf996dbe7229a7bb1d234c97aea97a252c5f7c89f8547b6d091db0f0e40","side":"right"},{"sibling":"55bcbd4da3e20d93931f7e58673f10232e81a5b1514d7396cb4b71e8f95788d0","side":"right"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":152106,"merkle_root":"ac5182c6f3dd09931f2a689df5f4be36df7b55e55bcf106e195671f5ed55fd8f","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260526T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-26T05:37:34Z","sig_algorithm":"ed25519","signature":"a401243fbd2c077d29a623c6ef616c78fbd5c8ce1930afa7af7165b2386e5a8cf15d5094983a1c962e71b27b911a3f3ce09c8cfab9393be2ce5ce8ee6513da06","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_8d23bba11597bfa94a290c508eadc520c5b8e29e6e6f55d9fbb44c653cee5616"}}