Geometric complexity theory for product-plus-power

TL;DR

The paper studies border Waring rank and Kumar complexity through a geometric lens by introducing the product-plus-power model and its degenerations. It proves a converse to Kumar's theorem, showing that for a homogeneous f either $ ext{underline WR}(f) \,\le\, ext{underline Kc}(f)$ or f is a product of linear forms, and it relates these notions via Newton identities. It then deborders the product-plus-power model, constructs a detailed GCT program with stabilizers, orbit closures, and multiplicity obstructions, and derives new obstructions grounded in the symmetries of the polynomials, with a striking link to the Alon-Tarsi conjecture. A key consequence is a novel representation-theoretic route to characterize the matrix multiplication exponent via $\,igl. \, ext{ω} = \inf\{\tau: 2n\times 2n\ matrices; M(n) = O(n^\tau)\}$, connected to $ ext{underline WR}$ and $ ext{underline Kc}$ of trace polynomials in the limit. Overall, the work advances the understanding of border vs non-border models, provides concrete GCT obstructions for product-plus-power, and ties complexity-theoretic questions to deep combinatorial symmetries.

Abstract

According to Kumar's recent surprising result (ToCT'20), a small border Waring rank implies that the polynomial can be approximated as a sum of a constant and a small product of linear polynomials. We prove the converse of Kumar's result and establish a tight connection between border Waring rank and the model of computation in Kumar's result. In this way, we obtain a new formulation of border Waring rank, up to a factor of the degree. We connect this new formulation to the orbit closure problem of the product-plus-power polynomial. We study this orbit closure from two directions: 1. We deborder this orbit closure and some related orbit closures, i.e., prove all points in the orbit closure have small non-border algebraic branching programs. 2. We fully implement the geometric complexity theory approach against the power sum by generalizing the ideas of Ikenmeyer-Kandasamy (STOC'20) to this new orbit closure. In this way, we obtain new multiplicity obstructions that are constructed from just the symmetries of the polynomials.

Theorems & Definitions (68)

• Theorem 1.1: Converse of Kumar's theorem
• Corollary 1.2
• Theorem 1.3: Debordering product-plus-power
• Theorem 1.4: New obstructions
• Remark 2.1
• Proposition 2.2: Newton Identities, see e.g. Macdon:SymmetricFunctions, Section I.2
• Lemma 2.3
• proof
• Proposition 2.4: kum20
• proof