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We prove that every EKP-homaloidal polynomial function of degree three is a relative invariant of a symmetric prehomogeneous vector space. This provides a complete proof of \\cite[Theorem 3.10, p.~39]{MR1890194}. With respect to the original argument of Etingof, Kazhdan and Polischuk our argument focuses more on prehomogeneous vector sp"},"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":"1011.5975","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-11-27T14:38:48Z","cross_cats_sorted":[],"title_canon_sha256":"e61e9de5fe54dfafd9493f7579a8e7156b0a7c936452a74ef6a6497c8e586a58","abstract_canon_sha256":"5e752ad8918878bfd3f7042dfd384de125090e6c16675563920a4e0c43b980f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:37.757083Z","signature_b64":"7Wk1dq/K2Ro40qMixKWlGvDvYGfVzw8K0A9rKYQqsHmlEIrrN40c2DBFIHBLf6UI/N8BTBe9X0n7HZemlJWQCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44fb3f4770440e314780398c972f9be1f486549f8a4267bcb1a24c185ca87900","last_reissued_at":"2026-05-18T04:34:37.756692Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:37.756692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Pierre-Emmanuel Chaput, Pietro Sabatino","submitted_at":"2010-11-27T14:38:48Z","abstract_excerpt":"In this paper we consider homaloidal polynomial functions $f$ such that their multiplicative Legendre transform $f_*$, defined as in \\cite[Section3.2]{MR1890194}, is again polynomial. Following Dolgachev \\cite{MR1786486}, we call such polynomials EKP-homaloidal. We prove that every EKP-homaloidal polynomial function of degree three is a relative invariant of a symmetric prehomogeneous vector space. This provides a complete proof of \\cite[Theorem 3.10, p.~39]{MR1890194}. 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