Degree-restricted strength decompositions and algebraic branching programs

We analyze Kumar's recent quadratic algebraic branching program size lower bound proof method (CCC 2017). We provide a refinement of this method and show examples in which the refined method gives a better lower bound than the original one. 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. Further, we use similar methods to improve the known lower bound methods for slice rank of polynomials. We give a sequence of polynomials for which the improved lower bound matches the known upper bound.