<p>We use the circle method to prove that a density 1 of elements in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3765_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_q[t]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are representable as a sum of three cubes of essentially minimal degree from <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3765_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_q[t]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, assuming the Ratios Conjecture and that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3765_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{char}\,}}(\mathbb {F}_q)>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>char</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting.</p>

Optimal sums of three cubes in $\mathbb {F}_q[t]$

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2025年5月23日

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Tim Browning,
Jakob Glas,
Victor Y. Wang

摘要

We use the circle method to prove that a density 1 of elements in $\mathbb {F}_q[t]$ $F_{q} [t]$ are representable as a sum of three cubes of essentially minimal degree from $\mathbb {F}_q[t]$ $F_{q} [t]$ , assuming the Ratios Conjecture and that ${{\,\textrm{char}\,}}(\mathbb {F}_q)>3$ $char (F_{q}) > 3$ . Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting.

Optimal sums of three cubes in \(\mathbb {F}_q[t]\)

摘要