Sum–product estimates for rational functions
Sum–product estimates for rational functions
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有理函数的和积估计
DOI：
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发表时间：
2010
期刊：
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通讯作者：
Jacob Tsimerman
中科院分区：
文献类型：
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作者：
B. Bukh;Jacob Tsimerman
我们建立了几个有限域上的和积估计，涉及多项式和有理函数。第一、|f（A）+f（A）+|AA|基本上大于|一|对于任意的多项式f？第二，给出了有理函数f和g的特征，其中|f（A）+f（A）+|g（A，A）|可以小到|一|大型|一|.第三，我们证明了在f上的温和条件下，|f（A，A）|基本上大于|一|，前提是|一|很大。我们还提出了一个猜想，一般的和产品的结果应该是什么。
We establish several sum–product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)+|AA| is substantially larger than |A| for an arbitrary polynomial f over ?p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)+|g(A, A)| can be as small as |A| for large |A|. Third, we show that under mild conditions on f, |f(A, A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what the general sum–product result should be.