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Codes Cryptogr."],"published-print":{"date-parts":[[2025,10]]},"abstract":"Abstract<\/jats:title>\n Given two irreducible conics C<\/jats:italic> and D<\/jats:italic> over a finite field \n \n $$\\mathbb {F}_q$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with q<\/jats:italic> odd, we show that there are \n \n $$q^2\/4+O(q^{3\/2})$$<\/jats:tex-math>\n \n \n \n q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \/<\/mml:mo>\n 4<\/mml:mn>\n +<\/mml:mo>\n O<\/mml:mi>\n \n (<\/mml:mo>\n \n q<\/mml:mi>\n \n 3<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> points P<\/jats:italic> in \n \n $$\\mathbb {P}^2(\\mathbb {F}_q)$$<\/jats:tex-math>\n \n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> such that P<\/jats:italic> is external to C<\/jats:italic> and internal to D<\/jats:italic>. This answers a question of Korchm\u00e1ros. We also prove the analogous result for higher-dimensional smooth quadric hypersurfaces in \n \n $$\\mathbb {P}^{n-1}$$<\/jats:tex-math>\n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with n<\/jats:italic> odd, where the answer is \n \n $$q^{n-1}\/4+O(q^{n-\\frac{3}{2}})$$<\/jats:tex-math>\n \n \n \n q<\/mml:mi>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \/<\/mml:mo>\n 4<\/mml:mn>\n +<\/mml:mo>\n O<\/mml:mi>\n \n (<\/mml:mo>\n \n q<\/mml:mi>\n \n n<\/mml:mi>\n -<\/mml:mo>\n \n 3<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10623-025-01687-9","type":"journal-article","created":{"date-parts":[[2025,7,6]],"date-time":"2025-07-06T07:35:52Z","timestamp":1751787352000},"page":"4461-4472","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mutual position of two smooth quadrics over finite fields"],"prefix":"10.1007","volume":"93","author":[{"given":"Shamil","family":"Asgarli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chi Hoi","family":"Yip","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,6]]},"reference":[{"issue":"4","key":"1687_CR1","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1515\/advgeom.2011.022","volume":"11","author":"V Abatangelo","year":"2011","unstructured":"Abatangelo V., Fisher J.C., Korchm\u00e1ros G., Larato B.: On the mutual position of two irreducible conics in $${\\rm PG}(2, q)$$, $$q$$ odd. 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