On the number of rational points of certain algebraic varieties 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 odd characteristic p with q elements ( <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>n</m:mi> </m:msup> </m:mrow> </m:math> {q=p^{n}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>∈</m:mo> <m:mi>ℕ</m:mi> </m:mrow> </m:math> {n\in\mathbb{N}} ) and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mo>*</m:mo> </m:msubsup> </m:math> {\mathbb{F}_{q}^{*}} represent the set of nonzero elements of <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}} . By making use of the Smith normal form of exponent matrices, we obtain an explicit formula for the number of rational points on the variety defined by the following system of equations 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}} : <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd /> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>r</m:mi> </m:munderover> <m:mrow> <m:msubsup> <m:mi>a</m:mi> <m:mi>i</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:msubsup> <m:mi>e</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> </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:msubsup> <m:mi>e</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> </m:msubsup> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mi>b</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd /> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:munderover> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:msup> <m:mi>i</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mi>r</m:mi> <m:mrow> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>-</m:mo> <m:msub> <m:mi>r</m:mi> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> </m:mrow> </m:munderover> <m:mrow> <m:msubsup> <m:mi>a</m:mi> <m:mrow> <m:msub> <m:mi>r</m:mi> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> <m:mo>+</m:mo> <m:msup> <m:mi>i</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:msubsup> <m:mi>e</m:mi> <m:mrow> <m:mrow> <m:msub> <m:mi>r</m:mi> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> <m:mo>+</m:mo> <m:msup> <m:mi>i</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> </m:msubsup> <m:mo>⁢</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:msub> <m:mi>n</m:mi> <m:mrow> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msubsup> <m:mi>e</m:mi> <m:mrow> <m:mrow> <m:msub> <m:mi>r</m:mi> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> <m:mo>+</m:mo> <m:msup> <m:mi>i</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mi>n</m:mi> <m:mrow> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> </m:msubsup> </m:mrow> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mi>b</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \left\{\begin{aligned} &amp;\displaystyle\sum_{i=1}^{r}a^{(1)}_{i}x_{1}^{e_{i1}^{(% 1)}}\cdots x_{n}^{e_{in}^{(1)}}=b_{1},\\ &amp;\displaystyle\sum^{t-1}_{j^{\prime}=0}\sum^{r_{j^{\prime}+1}-r_{j^{\prime}}}_% {i^{\prime}=1}a^{(2)}_{r_{j^{\prime}}+i^{\prime}}x_{1}^{e_{r_{j^{\prime}}+i^{% \prime},1}^{(2)}}\cdots x_{n_{{j^{\prime}}+1}}^{e_{r_{j^{\prime}}+i^{\prime},n% _{{j^{\prime}}+1}}^{(2)}}=b_{2},\end{aligned}\right.\vspace*{1mm} where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>b</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∈</m:mo> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:mrow> </m:math> {b_{i}\in\mathbb{F}_{q}} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> {i=1,2} ), <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mi>ℕ</m:mi> </m:mrow> </m:math> {t\in\mathbb{N}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>0</m:mn> <m:mo>=</m:mo> <m:msub> <m:mi>n</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>&lt;</m:mo> <m:msub> <m:mi>n</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>&lt;</m:mo> <m:msub> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>&lt;</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>&lt;</m:mo> <m:msub> <m:mi>n</m:mi> <m:mi>t</m:mi> </m:msub> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> 0=n_{0}&lt;n_{1}&lt;n_{2}&lt;\cdots&lt;n_{t},\vspace*{1mm} <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>n</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>&lt;</m:mo> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:msub> <m:mi>n</m:mi> <m:mi>k</m:mi> </m:msub> </m:mrow> </m:math> {n_{k-1}&lt;n\leq n_{k}} for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>k</m:mi> <m:mo>≤</m:mo> <m:mi>t</m:mi> </m:mrow> </m:math> {1\leq k\leq t} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>0</m:mn> <m:mo>=</m:mo> <m:msub> <m:mi>r</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>&lt;</m:mo> <m:msub> <m:mi>r</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>&lt;</m:mo> <m:msub> <m:mi>r</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>&lt;</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>&lt;</m:mo> <m:msub> <m:mi>r</m:mi> <m:mi>t</m:mi> </m:msub> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> 0=r_{0}&lt;r_{1}&lt;r_{2}&lt;\cdots&lt;r_{t},\vspace*{1mm} <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>a</m:mi> <m:mi>i</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <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> {a^{(1)}_{i}\in\mathbb{F}_{q}^{*}} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {i\in\{1,\ldots,r\}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>a</m:mi> <m:msup> <m:mi>i</m:mi> <m:mo>′</m:mo> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <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> {a^{(2)}_{i^{\prime}}\in\mathbb{F}_{q}^{*}} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>i</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>r</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {{i^{\prime}}\in\{1,\ldots,r_{t}\}} , and the exponent of each variable is a positive integer. This generalizes the results obtained previously by Wolfmann, Sun, Cao, and others. Our result also gives a partial answer to an open problem raised by Hu, Hong and Zhao [S. N. Hu, S. F. Hong and W. Zhao, The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 2015, 135–153].