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

Abstract

Let Fq{{\mathbb{F}}}_{q} be the finite field of characteristic pp and Fq*=Fq\{0}{{\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 Fq{{\mathbb{F}}}_{q}: a1x1e11…xm1e1m1+…+am1x1em1,1…xm1em1,m1=b1,am1+1x1em1+1,1…xm2em1+1,m2+…+am2x1em2,1…xm2em2,m2=b2\left\{\begin{array}{l}{a}_{1}{x}_{1}^{{e}_{11}}\ldots {x}_{{m}_{1}}^{{e}_{1{m}_{1}}}+\ldots +{a}_{{m}_{1}}{x}_{1}^{{e}_{{m}_{1},1}}\ldots {x}_{{m}_{1}}^{{e}_{{m}_{1},{m}_{1}}}={b}_{1},\\ {a}_{{m}_{1}+1}{x}_{1}^{{e}_{{m}_{1}+\mathrm{1,1}}}\ldots {x}_{{m}_{2}}^{{e}_{{m}_{1}+1,{m}_{2}}}+\ldots +{a}_{{m}_{2}}{x}_{1}^{{e}_{{m}_{2},1}}\ldots {x}_{{m}_{2}}^{{e}_{{m}_{2},{m}_{2}}}={b}_{2}\end{array}\right. and c1x1d11…xn1d1n1+…+cn1x1dn1,1…xn1dn1,n1=l1,cn1+1x1dn1+1,1…xn2dn1+1,n2+…+cn2x1dn2,1…xn2dn2,n2=l2,cn2+1x1dn2+1,1…xn3dn2+1,n3+…+cn3x1dn3,1…xn3dn3,n3=l3\left\{\begin{array}{l}{c}_{1}{x}_{1}^{{d}_{11}}\ldots {x}_{{n}_{1}}^{{d}_{1{n}_{1}}}+\ldots +{c}_{{n}_{1}}{x}_{1}^{{d}_{{n}_{1},1}}\ldots {x}_{{n}_{1}}^{{d}_{{n}_{1},{n}_{1}}}={l}_{1},\\ {c}_{{n}_{1}+1}{x}_{1}^{{d}_{{n}_{1}+\mathrm{1,1}}}\ldots {x}_{{n}_{2}}^{{d}_{{n}_{1}+1,{n}_{2}}}+\ldots +{c}_{{n}_{2}}{x}_{1}^{{d}_{{n}_{2},1}}\ldots {x}_{{n}_{2}}^{{d}_{{n}_{2},{n}_{2}}}={l}_{2},\\ {c}_{{n}_{2}+1}{x}_{1}^{{d}_{{n}_{2}+\mathrm{1,1}}}\ldots {x}_{{n}_{3}}^{{d}_{{n}_{2}+1,{n}_{3}}}+\ldots +{c}_{{n}_{3}}{x}_{1}^{{d}_{{n}_{3},1}}\ldots {x}_{{n}_{3}}^{{d}_{{n}_{3},{n}_{3}}}={l}_{3}\end{array}\right.\hspace{1.15em} when the determinants of exponent matrices are coprime to q−1q-1, where eij,di′j′∈Z+(the set of positive integers),ai,ci′∈Fq*,1≤i,j≤m2,1≤i′,j′≤n3,{e}_{ij},{d}_{i^{\prime} j^{\prime} }\in {{\mathbb{Z}}}^{+}\hspace{0.1em}\text{(the set of positive integers)}\hspace{0.1em},{a}_{i},{c}_{i^{\prime} }\in {{\mathbb{F}}}_{q}^{* },1\le i,j\le {m}_{2},1\le i^{\prime} ,j^{\prime} \le {n}_{3}, and b1,b2,l1,l2,l3∈Fq{b}_{1},{b}_{2},{l}_{1},{l}_{2},{l}_{3}\in {{\mathbb{F}}}_{q}. These formulas extend the theorem obtained by Wang and Sun (An explicit formula of solution of some special equations over a finite field, Chinese Ann. Math. Ser. A 26 (2005), 391–396, https://www.cqvip.com/doc/journal/977048790. (in Chinese)). Our results also give a partial answer to an open problem of Hu et al. raised in (The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015), 135–153, doi: https://doi.org/10.1016/j.jnt.2015.04.006).

Keywords

• finite field
• algebraic variety
• rational point
• smith normal form
• prime number
• 11c20
• 11a05
• 15b36