{"paper":{"title":"Simultaneous Diagonalization of Conics in $PG(2,q)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Katharina Kusejko","submitted_at":"2014-10-15T07:55:33Z","abstract_excerpt":"Consider two symmetric $3 \\times 3$ matrices $A$ and $B$ with entries in $GF(q)$, for $q=p^n$, $p$ an odd prime. The zero sets of $v^T Av$ and $v^T Bv$ can be viewed as (possibly degenerate) conics in the finite projective coordinate plane of order $q$. Using combinatorial properties of pencils of conics in $PG(2,q)$, we are able to tell when it is possible to find a nonsingular matrix $S$ with entries in $GF(q)$, such that $S^T A S$ and $S^T BS$ are both diagonal matrices. This is equivalent to the existence of a collineation mapping two given conics into conics with matrices in diagonal form"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3954","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}