Geometric complexity theory for product-plus-power
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.
Cite
@article{arxiv.2211.07055,
title = {Geometric complexity theory for product-plus-power},
author = {Pranjal Dutta and Fulvio Gesmundo and Christian Ikenmeyer and Gorav Jindal and Vladimir Lysikov},
journal= {arXiv preprint arXiv:2211.07055},
year = {2025}
}
Comments
This version (v3) has been accepted for publication at the "special issue on the topics of MEGA 2024" of the Journal of Symbolic Computation. Parts of v1/v2 have been published independently as "Fixed-parameter debordering of Waring rank" (DOI: 10.4230/LIPIcs.STACS.2024.30) and "Homogeneous Algebraic Complexity Theory and Algebraic Formulas" (DOI: 10.4230/LIPIcs.ITCS.2024.43)