The number of rational points of some classes of algebraic varieties over finite fields

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> {{\mathbb{F}}}_{q} be the finite field of characteristic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msubsup> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mo>\</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {{\mathbb{F}}}_{q}^{* }\left={{\mathbb{F}}}_{q}\backslash \left\{0\right\} . In this article, we use Smith normal form of exponent matrices to present exact formulas for the numbers of rational points on suitable affine algebraic varieties defined by the following systems of equations over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> {{\mathbb{F}}}_{q} : <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msubsup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mn>11</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mo>…</m:mo> <m:msubsup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:msub> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> </m:msubsup> <m:mo>+</m:mo> <m:mo>…</m:mo> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:msubsup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow>