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We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of $L_1,\\dots,L_s$. The input consists of generators for the homogeneous ideals $I_X, I_{D_i} \\subset \\mathbb{C}[x_0,\\ldots,x_r]$ defining $X$ and $D_i$."},"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":"1709.08674","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-09-25T18:55:37Z","cross_cats_sorted":[],"title_canon_sha256":"76ae10b89f05b98eca58df2d054c8782d6a53c1b2051569c1a2335d28a8d011d","abstract_canon_sha256":"b05c4e0532605d3c734e4b4833d95ea70c74fb81b4977e24dcb903d7776b7de5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:32.366979Z","signature_b64":"UmsZspJYJPCKWlZfkl01RHIdaejNB4k7tuoYhPw3tvU9274Bp+W4wW3QWIuCWHx0OcCCuk7n35V5nT7W5jUSCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95daab931c3234f8d81ae876c00d152fd05ee3ecf20a4f6e11340ef0e4a2a87c","last_reissued_at":"2026-05-18T00:32:32.366295Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:32.366295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Numerical Polar calculus and cohomology of line bundles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Chris Peterson, David Eklund, Sandra Di Rocco","submitted_at":"2017-09-25T18:55:37Z","abstract_excerpt":"Let $L_1,\\dots,L_s$ be line bundles on a smooth variety $X\\subset \\mathbb{P}^r$ and let $D_1,\\dots,D_s$ be divisors on $X$ such that $D_i$ represents $L_i$. We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of $L_1,\\dots,L_s$. 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