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Counting solutions of special linear equations over finite fields
Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2020-09-25 , DOI: 10.1016/j.ffa.2020.101759
Lucas Reis

Let q be a prime power, let Fq be the finite field with q elements and let d1,,dk be positive integers. In this note we explore the number of solutions (z1,,zk)Fqk of the equationL1(x1)++Lk(xk)=b, with the restrictions ziFqdi, where each Li(x) is a non zero polynomial of the form j=0miaijxqjFq[x] and bFq. We characterize the elements b for which the equation above has a solution and, in affirmative case, we determine the exact number of solutions. As an application of our main result, we obtain the cardinality of the sumseti=1kFqdi:={α1++αk|αiFqdi}. Our approach also allows us to solve another interesting problem, regarding the existence and number of elements in Fqn with prescribed traces over intermediate Fq-extensions of Fqn.



中文翻译:

有限域上特殊线性方程组的计数解

q为素数,令Fq是具有q个元素的有限域,并令d1个dķ是正整数。在本说明中,我们探讨了许多解决方案ž1个žķFqķ 等式的大号1个X1个++大号ķXķ=b 有限制 ž一世Fqd一世,每个 大号一世X 是形式为非零的多项式 Ĵ=0一世一种一世ĴXqĴFq[X]bFq。我们对元素b进行特征化,上面的方程式有一个解,并且在肯定的情况下,我们确定解的确切数目。作为我们主要结果的应用,我们获得了和集的基数一世=1个ķFqd一世={α1个++αķ|α一世Fqd一世} 我们的方法还使我们能够解决另一个有趣的问题,涉及元素中元素的存在和数量 Fqñ 在中间有规定的痕迹 Fq-扩展 Fqñ

更新日期:2020-09-26
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