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For a forest graph $G$, we derive an explicit formula for the localized $\\mathrm{v}$-number of the LSS ideal $L_G^{\\mathbb{K}}(d)$, denoted by $\\mathrm{v}_{\\mathfrak{p}_{\\emptyset}(G)}(L_G^{\\mathbb{K}}(d))$, for all $d \\geq 2$, where $\\mathbb{K}$ is an algebraically closed field. As a consequence, we prove 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":"2608.01199","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-08-02T12:30:33Z","cross_cats_sorted":[],"title_canon_sha256":"bce5118582a09d9a32b5a3d1c89d62a7b22b4c338ad0e5932d764fb315bb2df8","abstract_canon_sha256":"6e08d33c33e1f14f5fe84b4cf54e9011b811baf84f93ba896bde00c5f7ecfc7f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:01:55.306750Z","signature_b64":"Wn5+3vjnSP6+w204r5Wf3hNVBheeCUS/m1rRAowmMvof5eSC5r76zhrNsedWWeD122XyUrv9OMgaYRi+8vt+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50ae584c2583d936681220358d333db54733f792949a3bf5239fe9316c67072b","last_reissued_at":"2026-08-04T02:01:55.304598Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:01:55.304598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\mathrm{v}$-number of Lov\\'asz-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Emiliano Liwski, Manohar Kumar","submitted_at":"2026-08-02T12:30:33Z","abstract_excerpt":"In this paper, we introduce a new framework for computing certain localized $\\mathrm{v}$-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lov\\'asz-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph $G$, we derive an explicit formula for the localized $\\mathrm{v}$-number of the LSS ideal $L_G^{\\mathbb{K}}(d)$, denoted by $\\mathrm{v}_{\\mathfrak{p}_{\\emptyset}(G)}(L_G^{\\mathbb{K}}(d))$, for all $d \\geq 2$, where $\\mathbb{K}$ is an algebraically closed field. 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