Given two irreducible conics C and D over a finite field <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1687_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> with q odd, we show that there are <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1687_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </InlineMediaObject> <EquationSource Format="TEX">$q^2/4+O(q^{3/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mn>4</mn> <mo>+</mo> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> points P in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1687_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {P}^2(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that P is external to C and internal to D. This answers a question of Korchmáros. We also prove the analogous result for higher-dimensional smooth quadric hypersurfaces in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1687_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {P}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> with n odd, where the answer is <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1687_Article_IEq5.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="128" /> </InlineMediaObject> <EquationSource Format="TEX">$q^{n-1}/4+O(q^{n-\frac{3}{2}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>q</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">/</mo> <mn>4</mn> <mo>+</mo> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow> <mi>n</mi> <mo>-</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Mutual position of two smooth quadrics over finite fields

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2025年7月6日

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Shamil Asgarli,
Chi Hoi Yip

摘要

Given two irreducible conics C and D over a finite field $\mathbb {F}_q$ $F_{q}$ with q odd, we show that there are $q^2/4+O(q^{3/2})$ $q^{2} / 4 + O (q^{3 / 2})$ points P in $\mathbb {P}^2(\mathbb {F}_q)$ $P^{2} (F_{q})$ such that P is external to C and internal to D. This answers a question of Korchmáros. We also prove the analogous result for higher-dimensional smooth quadric hypersurfaces in $\mathbb {P}^{n-1}$ $P^{n - 1}$ with n odd, where the answer is $q^{n-1}/4+O(q^{n-\frac{3}{2}})$ $q^{n - 1} / 4 + O (q^{n - \frac{3}{2}})$ .