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For example if $F$ is a homogeneous non-expander polynomial then $F(x,y)=c(x+ay)^\\alpha$ or $F(x,y)=c(xy)^\\alpha .$ This is an extension of an earlier result of Elekes and R\\'onyai who described the structure of two-variate polynomials which are not expanders over the reals."},"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":"1212.3365","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-13T23:27:07Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"293950227f4e25cbc8922df9b5bfae5597ebeec8290ab026e5657e685e61cab7","abstract_canon_sha256":"e43f9163e735aa5cf0fc94a75d373f3ed7d4e4f0a083b03e91994e0d0b4277aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:24.865589Z","signature_b64":"tomvJuL8VMCTPBnl047nmMnFnt3WKrgr3P1OogG0nHzgeFd+e1HnV5+HTkatkcbXFI+fazOqCIPEkss171UTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8305ab2041d41863db92297f8b26bb5468d500260ff0ee2635f4f7f856a8d82a","last_reissued_at":"2026-05-18T03:38:24.865077Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:24.865077Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Expanding Polynomials over the rationals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Jozsef Solymosi","submitted_at":"2012-12-13T23:27:07Z","abstract_excerpt":"Let $F(x,y)$ be a polynomial over the rationals. 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