Degree-restricted strength decompositions and algebraic branching programs

TL;DR

This paper refines algebraic branching program lower bound techniques for the power sum polynomial, using intersection theory and Noether-Lefschetz conditions, and improves bounds for polynomial slice rank.

Contribution

It introduces a refined method for algebraic branching program lower bounds and applies it to specific polynomials, matching known upper bounds.

Findings

Refined lower bounds for algebraic branching programs.

Application of intersection theory to polynomial bounds.

Matching bounds for specific polynomial sequences.

Abstract

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.