| 摘要: |
| Fq[t]为含有q个元的有限域Fq上的多项式环.对N∈N，设GN为由Fq[t]中一切次数严格小于N的多项式所形成的集合.假定Fq的特征严格大于2，并且A⊆GN2.如果对任何d∈Fq[t]\{0}都有（d，d2）∉A-A={a-a'：a，a'∈A}.本文证明了|A|≤Cq2N log N/N，此处常数C只依赖于q.应用这个估计，本文把函数域中的Sárközy定理推广到了次数严格小于3的多项式的有限族的情形. |
| 关键词: Sárközy定理 函数域 Hardy-Littlewood圆法 |
| DOI： |
| 分类号:O156 |
| 基金项目:Supported by National Natural Science Foundation of China(11671271). |
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| A 2-DIMENSIONAL ANALOGUE OF SÁRKÖZY'S THEOREM IN FUNCTION FIELDS |
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LI Guo-quan, LIU Bao-qing, QIAN Kun, XU Gui-qiao
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School of Mathematics Science, Tianjin Normal University, Tianjin 300387, China
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| Abstract: |
| Let Fq[t] be the polynomial ring over the finite field Fq of q elements. For N∈N, let GN be the set of all polynomials in Fq[t] of degree less than N. Suppose that the characteristic of Fq is greater than 2 and A⊆GN2. If (d,d2)∉A-A={a -a':a,a'∈A} for any d∈Fq[t]\{0}, we prove that|A| ≤ Cq2N log N/N, where the constant C depends only on q. By using this estimate, we extend Sárközy's theorem in function fields to the case of a finite family of polynomials of degree less than 3. |
| Key words: Sárközy's theorem function fields Hardy-Littlewood circle method |
