We analyze Kumar's recent quadratic algebraic branching program size lower bound proof method (CCC 2017) for the power sum polynomial. We present a refinement of this method that gives better bounds in some cases. The lower bound relies on Noether-Lefschetz type conditions on the hypersurface defined by the homogeneous polynomial. In the explicit example that we provide, the lower bound is proved resorting to classical intersection theory. Furthermore, we use similar methods to improve the known lower bound methods for slice rank of polynomials. We consider a sequence of polynomials that have been studied before by Shioda and show that for these polynomials the improved lower bound matches the known upper bound.
翻译:我们分析了 Kumar 最近的四边代数分解程序对功率和多元度的低约束度验证方法( CCC 2017) 的大小。 我们展示了该方法的改进, 在某些情况中提供了更好的界限。 较低约束依赖于由同质多元度定义的超表层上的Noether- Lefschetz 类型条件 。 在我们提供的明显例子中, 较低约束被证明采用了经典交叉理论 。 此外, 我们使用类似方法来改进已知的多元度级切片的较低约束方法 。 我们考虑了 Shioda 之前研究过的多元度序列, 并显示对这些多元度中, 改进后的下约束与已知的上约束相匹配 。