On the number of zeros of diagonal cubic forms over finite fields

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> {\mathbb{F}_{q}} be the finite field of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>p</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> {q=p^{k}} elements with p being a prime and let k be a positive integer. For any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:mrow> </m:math> {y,z\in\mathbb{F}_{q}} , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {N_{s}(z)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>T</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {T_{s}(y)} denote the numbers of zeros of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mn>3</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mi>s</m:mi> <m:mn>3</m:mn> </m:msubsup> </m:mrow> <m:mo>=</m:mo> <m:mi>z</m:mi> </m:mrow> </m:math> {x_{1}^{3}+\cdots+x_{s}^{3}=z} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mn>3</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>+</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mrow> <m:mi>s</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mn>3</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mi>s</m:mi> <m:mn>3</m:mn> </m:msubsup> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {x_{1}^{3}+\cdots+x_{s-1}^{3}+yx_{s}^{3}=0} , respectively. Gauss proved that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>=</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> {q=p} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≡</m:mo> <m:mrow> <m:mpadded width="+3.3pt"> <m:mn>1</m:mn> </m:mpadded> <m:mspace width="veryverythickmathspace" /> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi /> <m:mo lspace="2.5pt" rspace="5.8pt">mod</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {p\equiv 1~{}(\bmod~{}3)} and y is non-cubic, then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>T</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mn>9</m:mn> <m:mo>⁢</m:mo> <m:mi>d</m:mi> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> T_{3}(y)=p^{2}+\frac{1}{2}(p-1)(-c+9d), where c and d are uniquely determined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>c</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:mrow> <m:mn>27</m:mn> <m:mo>⁢</m:mo> <m:msup> <m:mi>d</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:mrow> </m:math> {4p=c^{2}+27d^{2}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>c</m:mi> <m:mo>≡</m:mo> <m:mrow> <m:mpadded width="+3.3pt"> <m:mn>1</m:mn> </m:mpadded> <m:mspace width="veryverythickmathspace" /> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi /> <m:mo lspace="2.5pt" rspace="5.8pt">mod</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {c\equiv 1~{}(\bmod~{}3)} except for the sign of d . In 1978, Chowla, Cowles and Cowles determined the sign of d for the case of 2 being a non-cubic element of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{F}_{p}} . But the sign problem is kept open for the remaining case of 2 being cubic in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{F}_{p}} . In this paper, we solve this sign problem by determining the sign of d when 2 is cubic in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{F}_{p}} . Furthermore, we show that the generating functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∑</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:msubsup> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>s</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {\sum_{s=1}^{\infty}N_{s}(z)x^{s}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∑</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:msubsup> <m:mrow> <m:msub> <m:mi>T</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>s</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {\sum_{s=1}^{\infty}T_{s}(y)x^{s}} are rational functions for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>z</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mo>*</m:mo> </m:msubsup> <m:mo>:=</m:mo> <m:mrow> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>∖</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {z,y\in\mathbb{F}_{q}^{*}:=\mathbb{F}_{q}\setminus\{0\}} with y being non-cubic over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> {\mathbb{F}_{q}} , and we also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.