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By normalizing the critical-point equations by a reference component, for $k\\ge2$ we get the direct Pfaffian bound \\[ U_{\\mathrm{het}}(d,k)=2^{\\,d+\\binom{k-1}{2}}\\left(d+2\\min(d,k-1)+1\\right)^{k-1}. \\] For the same parameter range, an exact elimination augmented by an algebraic reciprocal variable gives the alternative bound \\[ U_{\\mathrm{aug}}(d,k)= 2^{\\binom{k-1}{2}}(d+1","authors_text":"Hien Duy Nguyen","cross_cats":["math.CO","stat.TH"],"headline":"Gaussian mixture densities with k components have at most floor of (min of two algebraic bounds plus one) divided by two modes when the modal set is finite.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-05-15T02:02:36Z","title":"Bounds on the Number of Modes of a Gaussian Mixture Density"},"references":{"count":17,"internal_anchors":0,"resolved_work":17,"sample":[{"cited_arxiv_id":"","doi":"10.1093/imaiai/iaz013","is_internal_anchor":false,"ref_index":1,"title":"Alexandrovich, G., Holzmann, H., and Ray, S. 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Complexity of computations with Pfaffian and Noetherian functions. InNormal Forms, Bifurcations and Finiteness Problems in Differential Equations (pp. 211–250).","work_id":"e9263ee7-2412-4d7a-8268-9487722393e9","year":2004},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Kabata, Y., Matsumoto, H., Uchida, S., and Ueki, M. (2025). Singularities in bivariate normal mixtures.Information Geometry, 8, 343–357","work_id":"cc2fa01a-b997-4c1b-8ac6-f4030b4096f9","year":2025}],"snapshot_sha256":"b0bfa2899e3d9c9e0d8379c55651382760fdf4e6883de3d2c651c7e78e66996a"},"source":{"id":"2605.15531","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T14:31:00.102044Z","id":"06f58249-98d3-4191-b4e7-04975a0fac47","model_set":{"reader":"grok-4.3"},"one_line_summary":"Explicit upper bounds on nondegenerate critical points of k-component Gaussian mixture densities are given via Pfaffian and algebraic elimination methods, with homoscedastic simplifications and combinatorial lower bounds.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Gaussian mixture densities with k components have at most floor of (min of two algebraic bounds plus one) divided by two modes when the modal set is finite.","strongest_claim":"For k≥2, the direct Pfaffian bound is U_het(d,k)=2^{d+binom(k-1,2)}(d+2 min(d,k-1)+1)^{k-1}, with the best critical-point bound being the minimum of this and the augmented bound, and the finite-mode bound improved by Morse theory to floor((min{U_het,U_aug}+1)/2).","weakest_assumption":"The critical-point equations can be normalized by a reference component without loss of generality for k≥2, and the Morse-theoretic argument applies directly to improve the finite-mode upper bound when the modal set is finite."}},"verdict_id":"06f58249-98d3-4191-b4e7-04975a0fac47"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:272002c0c0fc04da0e427281b5f521010003b0fc26616c110fbae266e4408ef1","target":"record","created_at":"2026-05-20T00:01:03Z","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":"04cda0b859209116bf667dc4c05f72f30dc4eecc8e5058e89f1f0ddc84480ced","cross_cats_sorted":["math.CO","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-05-15T02:02:36Z","title_canon_sha256":"d84c2848a73d8cfd66c1a6b9dc5249c9d1964bf7bbc5e217f37f93f69b68e362"},"schema_version":"1.0","source":{"id":"2605.15531","kind":"arxiv","version":1}},"canonical_sha256":"980ebe5908387fb28703615a94f816ec4809a42ac93454b5d985e2073d297d00","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"980ebe5908387fb28703615a94f816ec4809a42ac93454b5d985e2073d297d00","first_computed_at":"2026-05-20T00:01:03.687197Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:01:03.687197Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dajvw72yHvlgsXs08opo3uTki+0GACfV4QhHEiuPvctw9xucNkZl+kxTws7Wm3PKCDnRYP/0NMF4ea9lS1rxDg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:01:03.688019Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.15531","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:272002c0c0fc04da0e427281b5f521010003b0fc26616c110fbae266e4408ef1","sha256:075d94d84737f552957f53d08dd6e8d1f3178f581f4a005aa8a5fbcaa30ab1ad"],"state_sha256":"73f47de8594a35e031c638f2bed3595a90bfa0cf5ac80cc9f335ccb583df5c16"}