论文标题
多项式方程模型素数
Polynomial equations modulo prime numbers
论文作者
论文摘要
我们考虑具有整数系数的多项式方程或多项式方程系统，Modulo Prime数字$ P $。我们提供基于计数方法的基本方法。结果是解决方案数量$ p $的Lang-Weil下限的弱形式，仅与Lang-Weil不同，渐近$ P^ε$乘数因子。我们的第二个贡献是对单个方程式的情况减少引理，我们用来将结果扩展到方程系统。我们进一步展示了如何使用此减少来证明品种的完整lang-weil估计值，假设它是用于Hypersurfaces的，则在错误项中使用经典程度的变体中的版本。
We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound for the number of solutions modulo $p$, only differing from Lang-Weil by an asymptotic $p^ε$ multiplicative factor. Our second contribution is a reduction lemma to the case of a single equation which we use to extend our results to systems of equations. We show further how to use this reduction to prove the full Lang-Weil estimate for varieties, assuming it for hypersurfaces, in a version using a variant of the classical degree in the error term.
