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We prove that the average number of isolated common zeros of $f_1 \\in V_1, \\ldots, f_n \\in V_n$ is equal to the mixed symplectic volume of these Finsler ellipsoids. If $X$ is a homogeneous space of a compact Lie group and all vector spaces $V_i$ and their Euclidean metrics are invariant, then the average numbers of zeros satisfy "},"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":"1802.02741","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-02-08T08:14:43Z","cross_cats_sorted":[],"title_canon_sha256":"e7e763a0dccadf6f37a8643403f44807fd54b52155b4ba30126cf9bde30ee6f7","abstract_canon_sha256":"9f5e85486d11eb39dbc60d3fd5667953ce4d1d146315bf69a3da0c4184e24410"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:17.653563Z","signature_b64":"LY51LWI1B1t3ungQUdaoQeKVlejAgqlD8OVmP2lQ4jJiAQerC7m+G+2t0a+ya11p4WQwi5XHBn6EO6ttdTtIDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f5835a06a6966572c389669102e4c4ade8fa7740767779b64bf8ffc5edfe671","last_reissued_at":"2026-05-17T23:59:17.653047Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:17.653047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Average number of zeros and mixed symplectic volume of Finsler sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Boris Kazarnovskii, Dmitri Akhiezer","submitted_at":"2018-02-08T08:14:43Z","abstract_excerpt":"Let $X$ be an $n$-dimensional manifold and $V_1, \\ldots, V_n \\subset C^\\infty(X, \\mathbb R)$ finite-dimensional vector spaces with Euclidean metric. 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