On the number of zeros of diagonal quartic 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>m</m:mi> </m:msup> <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>4</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {q=p^{m}\equiv 1~{}(\bmod~{}4)} elements with p being an odd prime and m being a positive integer. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>c</m:mi> <m:mo>,</m:mo> <m:mi>y</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> {c,y\in\mathbb{F}_{q}} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mo>*</m:mo> </m:msubsup> </m:mrow> </m:math> {y\in\mathbb{F}_{q}^{*}} non-quartic, let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>c</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {N_{n}(c)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mi>n</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> {M_{n}(y)} be 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>4</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>n</m:mi> <m:mn>4</m:mn> </m:msubsup> </m:mrow> <m:mo>=</m:mo> <m:mi>c</m:mi> </m:mrow> </m:math> {x_{1}^{4}+\cdots+x_{n}^{4}=c} 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>4</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>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mn>4</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>n</m:mi> <m:mn>4</m:mn> </m:msubsup> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {x_{1}^{4}+\cdots+x_{n-1}^{4}+yx_{n}^{4}=0} , respectively. In 1979, Myerson used Gauss sums and exponential sums to show that the generating function <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>n</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>n</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:mo>⁢</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {\sum_{n=1}^{\infty}N_{n}(0)x^{n}} is a rational function in x and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:munderover> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>c</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mtext>and</m:mtext> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:munderover> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </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>n</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mrow> </m:math> \sum_{n=1}^{\infty}N_{n}(c)x^{n}\quad\text{and}\quad\sum_{n=1}^{\infty}M_{n+1}% (y)x^{n} are rational functions in x . We also obtain the explicit expressions of these generating functions. Our result extends Myerson’s theorem gotten in 1979.