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In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for each $1\\leq i\\leq n$ the system of Diophantine equations \\begin{equation*}\n  \\sigma_{i}(\\bar{X}_{2n})=a, \\quad \\sigma_{2n-i}(\\bar{X}_{2n})=b, \\quad \\sigma_{2n}(\\bar{X}_{2n})=c \\end{equation*} has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems invo"},"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":"1305.6241","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-27T14:39:55Z","cross_cats_sorted":[],"title_canon_sha256":"0e57bbb08192bab9701d5f62878c98ead8c9520ef80808e48591c946e5a7d9f0","abstract_canon_sha256":"62005eaf134865bf4c20e7fc8d4713ed1baf8a16b8e7cb76c56d92f7d606a7c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:24:51.786302Z","signature_b64":"1BMREx6uC1uHuqKa5lxjIEE3sBtvfM4lJUkCEmtWcypMItlXtRcp3ag0SVX05eU/hEmXfMO6lyY9Xrxl5km8CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8397b0300c014ca0536aea24e3241719d1e2e95ce3e2c331684bd55c80a11ccd","last_reissued_at":"2026-05-18T03:24:51.785736Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:24:51.785736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on Diophantine systems involving three symmetric polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maciej Ulas","submitted_at":"2013-05-27T14:39:55Z","abstract_excerpt":"Let $\\bar{X}_{n}=(x_{1},\\ldots,x_{n})$ and $\\sigma_{i}(\\bar{X}_{n})=\\sum x_{k_{1}}\\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for each $1\\leq i\\leq n$ the system of Diophantine equations \\begin{equation*}\n  \\sigma_{i}(\\bar{X}_{2n})=a, \\quad \\sigma_{2n-i}(\\bar{X}_{2n})=b, \\quad \\sigma_{2n}(\\bar{X}_{2n})=c \\end{equation*} has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. 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