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These conditions are well-known in the realm of bifurcation theory and commonly accepted as \"generic\". Here we show that under the transversality assumptions the sections $C(t)=\\{x:f(x,t)=0\\}$ of the zero set of $f$ are discrete for every $t\\in (0,T)$ and we discuss a somehow explicit family of perturbations of $f$ along which transversality holds up to a residual set.\n  The application of these results to the case when $f$ is the $X$-differe","authors_text":"Giuseppe Savar\\'e, Riccarda Rossi, Virginia Agostiniani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-10-17T20:28:07Z","title":"On the transversality conditions and their genericity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4846","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a57f457d50693f3b5de233b92b198b52ad9d3103fa98c29bf3dd02af8d1a6ae9","target":"record","created_at":"2026-05-18T03:10:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b95d1d3e3c5bfa2c9a71b3e7d100521adf2e2a0c38f6dd9f73b3ac91b85d3cf9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-10-17T20:28:07Z","title_canon_sha256":"645fcb1e178c1ba6e5a55132af2e66d17213f128c9eff1498fd40af5fb5cddb0"},"schema_version":"1.0","source":{"id":"1310.4846","kind":"arxiv","version":1}},"canonical_sha256":"4e47053fde734e6eff61838217a04d5bd7142973fd9f47470d64c8babc523888","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e47053fde734e6eff61838217a04d5bd7142973fd9f47470d64c8babc523888","first_computed_at":"2026-05-18T03:10:02.521283Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:10:02.521283Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pZDuSpGuA8HdYVQUfmJ6+Keo0lI6ntmJdJdr+2QLUPIQM01rgJY6KYtK7uk+u1yyKBMN4tQH86Qb6CVf5HeqAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:10:02.522158Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.4846","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a57f457d50693f3b5de233b92b198b52ad9d3103fa98c29bf3dd02af8d1a6ae9","sha256:879a3a3842c589d2f9e6be880979e34563d1872d45d2fbb5de13e45a85c0f809"],"state_sha256":"9b597049ce401ff07043d6de3300d27c5310a52bf3d86abb4deb73c7fef47c0a"}