We study the sum–product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum–product bounds that depend on these parameters. For the dual numbers we expose a range where the minimum value of $max\{|A+A|,\left|AA\right|\}$ is neither close to $\left|A\right|$ nor to ${\left|A\right|}^2$.

To obtain our main sum–product bound, we extend Elekes’ sum–product technique that relies on point–line incidences. Our extension is significantly more involved than the original proof, and in some sense runs the original technique a few times in a bootstrapping manner. We also study point–line incidences in the dual plane and in the double plane, developing analogs of the Szemerédi–Trotter theorem. As in the case of the sum–product problem, it turns out that the dual and double variants behave differently than the complex and real ones.

我们研究平面超复数的和积问题：对偶数和对偶数。这些数字系统类似于复数，但是事实证明它们具有非常不同的组合行为。我们确定控制这些问题行为的参数，并得出依赖于这些参数的总和乘积界限。对于双数，我们公开了一个范围，其中最小值为$最高\{|\mathrm{一种}+\mathrm{一种}|，\left|\mathrm{一种}\mathrm{一种}\right|\}$ 都不接近 $\left|\mathrm{一种}\right|$ 也不去 ${\left|\mathrm{一种}\right|}^2$。

为了获得我们的主要和积界限，我们扩展了Elekes的求和积技术，该技术依赖于点线关联。我们的扩展比原始证明要复杂得多，并且在某种意义上以自举方式运行了原始技术几次。我们还研究了Szemerédi-Trotter定理的类似物，研究了双平面和双平面中的点线入射。就像和积问题一样，事实证明，对偶和对偶变体的行为不同于复杂和实在变体。