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- W2963752859 abstract "Let $${mathbb {F}}_q$$ denote the finite field of odd characteristic p with q elements ( $$q=p^{n},nin {mathbb {N}} $$ ) and $${mathbb {F}}_q^*$$ represent the nonzero elements of $${mathbb {F}}_{q}$$ . In this paper, by using the Smith normal form we give a formula for the number of rational points of the algebraic varieties defined by the following system of equations over $${mathbb {F}}_{q}$$ : $$begin{aligned} left{ begin{array}{ll} sum limits _{i=1}^{r_1}a_{1i}x_1^{e^{(1)}_{i1}} ldots x_{n_1}^{e^{(1)}_{i,n_1}} +sum limits _{i=r_1+1}^{r_2}a_{1i}x_1^{e^{(1)}_{i1}} ldots x_{n_2}^{e^{(1)}_{i,n_2}}-b_1=0, sum limits _{j=1}^{r_3}a_{2j}x_1^{e^{(2)}_{j1}} ldots x_{n_3}^{e^{(2)}_{j,n_3}} +sum limits _{j=r_3+1}^{r_4}a_{2j}x_1^{e^{(2)}_{j1}} ldots x_{n_4}^{e^{(2)}_{j,n_4}}-b_2=0, end{array}right. end{aligned}$$ where the integers $$1le r_1<r_2$$ , $$1le r_3<r_4$$ , $$1le n_1<n_2$$ , $$1le n_3<n_4$$ , $$n_1le n_3$$ , $$b_1, b_2in mathbb {F}_{q}$$ , $$a_{1i}in mathbb {F}_{q}^{*}(1le ile r_2)$$ , $$a_{2j}in mathbb {F}_{q}^{*}(1le jle r_4)$$ and the exponent of each variable is positive integer. Our result provides a partial answer to an open problem raised in Hu et al. (J Number Theory 156:135–153, 2015)." @default.
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- W2963752859 date "2019-01-23" @default.
- W2963752859 modified "2023-09-24" @default.
- W2963752859 title "Counting Rational Points of an Algebraic Variety over Finite Fields" @default.
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- W2963752859 doi "https://doi.org/10.1007/s00025-019-0962-6" @default.
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