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This result is a generalization of a result of Cimpri\\vc, Helton, McCullough, and the author.\n  In the free left $\\RR\\axs$-module $\\RR^{1 \\times \\ell}\\axs$ we introduce notions of the (noncommutative) zero set of a left $\\RR\\axs$-submodule and of a real left $\\RR\\axs$-submodule. We prove that every element from $\\RR^{1 \\times \\ell}\\axs$ whose zero set contains the intersection of zero sets of elements from a finite subset $S \\subset \\RR^{1 \\times \\e"},"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":"1305.0799","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-05-03T18:39:14Z","cross_cats_sorted":[],"title_canon_sha256":"1007c42882917ae6d9cb99b2ff85c76d1e60c2e863ddabbe4ea6cdb8721bceb9","abstract_canon_sha256":"bbeeb5150615f430ebae8896c528b6fbf003b09e4d5a3f666df426809a08fb83"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:32.872026Z","signature_b64":"HXqeWBteVirk1WlckNAP5B93erUjp6H8yU6r/qboj7cQ4w0F2CxmvqPYarmv5P3/4rXiyB/VaspgIlxfDyIuDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbbc78e3b8451544892e1f6565b0270eb12d9d881a0d383692675995ea7803df","last_reissued_at":"2026-05-18T03:26:32.871582Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:32.871582Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Real Nullstellensatz for Matrices of Non-Commutative Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Christopher S. 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