Exact solutions to Waring's problem for finite fields
Exact solutions to Waring's problem for finite fields
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DOI：
10.4064/aa141-2-3
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发表时间：
2008-10
期刊：
影响因子：
0.7
通讯作者：
Arne Winterhof;C. V. D. Woestijne
中科院分区：
文献类型：
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作者：
Arne Winterhof;C. V. D. Woestijne
， xi∈Fq，即为Fq元素的k次幂的和。然后，我们可以将Waring函数g（k, q）定义为将Fq的所有元素表示为k次幂的和所需的最大求和数。我们注意到，通过一个简单的论证，我们有g(k, q) = g(k, q)，其中k = gcd（k, q−1）。因此，从现在开始，我们假设k除q - 1。一些作者已经建立了g（k, q）值的界限，对于参数k和q的各种选择，在[8]中给出了一个综述。对于指数k比q小的情况，有很强的结果。例如，当2≤k < q + 1时，通过直接应用有限域上变异点数的Weil界可得g(k, q) = 2[6,7,8]。在本文中，我们将研究指数k比q大的情况，我们不仅会得到一个边界，而且会得到两个无限族对（k, q）的g（k, q）的确切值。我们的主要结果如下。
with xi ∈ Fq, i.e., as a sum of kth powers of elements of Fq. We can then define the Waring function g(k, q) as the maximal number of summands needed to express all elements of Fq as sums of kth powers. We note that, by an easy argument, we have g(k, q) = g(k, q), where k = gcd(k, q − 1). Hence, we will assume from now on that k divides q − 1. Several authors have established bounds on the value of g(k, q) for various choices of the parameters k and q – a survey is given in [8]. For the cases where the exponent k is small compared to q, there are strong results. For example, whenever 2 ≤ k < q + 1, it follows that g(k, q) = 2 by a direct application of the Weil bound for the number of points on varieties over finite fields [6, 7, 8]. In this paper, we will look at the cases where the exponent k is large compared to q, and we will obtain not only a bound, but the exact value of g(k, q) for two infinite families of pairs (k, q). Our main results are the following.