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For a finite group $G$ denote by $\\delta(G)$ and $\\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\\kk$ and any $v\\in V^{G}\\setminus\\{0\\}$ or $v\\in V\\setminus\\{0\\}$ respectively, there exists a homogeneous invariant $f\\in\\kk[V]^{G}$ of positive degree at most $d$ such that $f(v)\\ne 0$. Let $P$ be a Sylow-$p$-subgroup of $G$ (which we take to be trivial if the group order is not divisble by $p$). We show that $\\delta(G)=|P|$. If $N_{G}(P)/P$ is cyclic, we show $\\sigma(G)\\ge|N_{G}(P)|$"},"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":"1308.0991","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-08-05T14:29:22Z","cross_cats_sorted":[],"title_canon_sha256":"a83fbbc734b335840ed6b22135075e2b52024e60902cbfc13dd97b8cf9ea33d9","abstract_canon_sha256":"d386014cea478117197953d59130b6234dfa113ca16dc78e36373e93afa32c3a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:09.998390Z","signature_b64":"nNaAA5boNo0mRoguNeSdLfkgp0kIsrfdfP6YXzmFsh4dGba1+nh1ud9lxORcuP9pZhs391PMOPq+vrQLnTVQAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d55fc96e4f0f359732165b0f4e6ff00cac6a4f0203d7fcda2b1c4788c4d0a261","last_reissued_at":"2026-05-18T02:49:09.997954Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:09.997954Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero-separating invariants for finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Jonathan Elmer, Martin Kohls","submitted_at":"2013-08-05T14:29:22Z","abstract_excerpt":"We fix a field $\\kk$ of characteristic $p$. For a finite group $G$ denote by $\\delta(G)$ and $\\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\\kk$ and any $v\\in V^{G}\\setminus\\{0\\}$ or $v\\in V\\setminus\\{0\\}$ respectively, there exists a homogeneous invariant $f\\in\\kk[V]^{G}$ of positive degree at most $d$ such that $f(v)\\ne 0$. Let $P$ be a Sylow-$p$-subgroup of $G$ (which we take to be trivial if the group order is not divisble by $p$). We show that $\\delta(G)=|P|$. If $N_{G}(P)/P$ is cyclic, we show $\\sigma(G)\\ge|N_{G}(P)|$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0991","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1308.0991","created_at":"2026-05-18T02:49:09.998026+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.0991v1","created_at":"2026-05-18T02:49:09.998026+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.0991","created_at":"2026-05-18T02:49:09.998026+00:00"},{"alias_kind":"pith_short_12","alias_value":"2VP4S3SPB42Z","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_16","alias_value":"2VP4S3SPB42ZOMQW","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_8","alias_value":"2VP4S3SP","created_at":"2026-05-18T12:27:32.513160+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS","json":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS.json","graph_json":"https://pith.science/api/pith-number/2VP4S3SPB42ZOMQWLMHU437QBS/graph.json","events_json":"https://pith.science/api/pith-number/2VP4S3SPB42ZOMQWLMHU437QBS/events.json","paper":"https://pith.science/paper/2VP4S3SP"},"agent_actions":{"view_html":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS","download_json":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS.json","view_paper":"https://pith.science/paper/2VP4S3SP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.0991&json=true","fetch_graph":"https://pith.science/api/pith-number/2VP4S3SPB42ZOMQWLMHU437QBS/graph.json","fetch_events":"https://pith.science/api/pith-number/2VP4S3SPB42ZOMQWLMHU437QBS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS/action/storage_attestation","attest_author":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS/action/author_attestation","sign_citation":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS/action/citation_signature","submit_replication":"https://pith.science/pith/2VP4S3SPB42ZOMQWLMHU437QBS/action/replication_record"}},"created_at":"2026-05-18T02:49:09.998026+00:00","updated_at":"2026-05-18T02:49:09.998026+00:00"}