{"paper":{"title":"Scalar relative differential invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The algebra of relative differential invariants becomes finitely generated after localization at a finite set of them.","cross_cats":["math.RA"],"primary_cat":"math.DG","authors_text":"Boris Kruglikov, Eivind Schneider","submitted_at":"2026-04-16T18:48:21Z","abstract_excerpt":"Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their (differential) algebra and demonstrate both positive and negative results in this respect under various setups. As in the algebraic case, the algebra of polynomial differential invariants is not finitely generated. However we show that after localization on a finite set of relative invariants the differential algebra becomes finitely generated. We also inve"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We address the fundamental problem of finite generation of their (differential) algebra and demonstrate both positive and negative results in this respect under various setups. ... after localization on a finite set of relative invariants the differential algebra becomes finitely generated. We also investigate the weights of rational relative differential invariants and bound their order.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The various setups for geometric structures admit well-defined relative differential invariants to which the algebraic finite-generation techniques can be extended.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Relative differential invariant algebras are not finitely generated in general but become so after localization, with order bounds for rational invariants.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The algebra of relative differential invariants becomes finitely generated after localization at a finite set of them.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"be3a31d55e4f5db2319568cf34d02bfdfaf2b9616d9c52467c6c249b6e10e81a"},"source":{"id":"2604.15473","kind":"arxiv","version":2},"verdict":{"id":"d1156a03-7d99-49b0-8509-82746bfacc3f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T09:39:08.842495Z","strongest_claim":"We address the fundamental problem of finite generation of their (differential) algebra and demonstrate both positive and negative results in this respect under various setups. ... after localization on a finite set of relative invariants the differential algebra becomes finitely generated. We also investigate the weights of rational relative differential invariants and bound their order.","one_line_summary":"Relative differential invariant algebras are not finitely generated in general but become so after localization, with order bounds for rational invariants.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The various setups for geometric structures admit well-defined relative differential invariants to which the algebraic finite-generation techniques can be extended.","pith_extraction_headline":"The algebra of relative differential invariants becomes finitely generated after localization at a finite set of them."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.15473/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}