论文标题
计算对角线二聚体方程的单色溶液
Counting monochromatic solutions to diagonal Diophantine equations
论文作者
论文摘要
我们展示了如何适应强壮的 - 小木圆方法将单色溶液计算为对角线溶液方程。这提供了最佳的下限，直至绝对常数。通过设置等于线性形式的对角线二次形式获得的方程式说明了该方法。结果，我们确定了该方程是常规分区时的代数标准。我们的方法涉及离散的谐波分析，并且需要许多“混合”限制估计值，这可能具有独立利益。
We show how to adapt the Hardy--Littlewood circle method to count monochromatic solutions to diagonal Diophantine equations. This delivers a lower bound which is optimal up to absolute constants. The method is illustrated on equations obtained by setting a diagonal quadratic form equal to a linear form. As a consequence, we determine an algebraic criterion for when such equations are partition regular. Our methods involve discrete harmonic analysis and require a number of `mixed' restriction estimates, which may be of independent interest.
