Problem
Research questions and friction points this paper is trying to address.
Proves converse of Kumar's border Waring rank result
Connects border Waring rank to orbit closure problem
Deborders orbit closure and implements GCT approach
De-bordering and Geometric Complexity Theory for Waring rank and related models
🤖 AI Summary
This paper addresses the exact characterization of border Waring rank in algebraic complexity theory, aiming to bridge the theoretical gap between approximate and exact computation. Methodologically, it integrates geometric complexity theory (GCT), invariant theory, representation theory, and orbit closure analysis. The contributions include: (i) the first exponential upper bound on border Waring rank—linear in the number of variables; (ii) a converse equivalence characterization of Kumar’s theorem; (iii) a novel closed-form description of the orbit closure of product-plus-power polynomials; and (iv) the first GCT multiplicative barrier constructed solely from symmetry considerations and branching rules. Collectively, these results enable a complete GCT attack on the power sum polynomial, extend the Ikenmeyer–Kandasamy framework to new orbit classes, and uncover the homogeneous essence of the Ben-Or–Cleve formula in the matrix setting.
📝 Abstract
De-bordering is the task of proving that a border complexity measure is bounded from below, by a non-border complexity measure. This task is at the heart of understanding the difference between Valiant's determinant vs permanent conjecture, and Mulmuley and Sohoni's Geometric Complexity Theory (GCT) approach to settle the P
eq NP conjecture. Currently, very few de-bordering results are known. In this work, we study the question of de-bordering the border Waring rank of polynomials. Waring and border Waring rank are very well studied measures, in the context of invariant theory, algebraic geometry and matrix multiplication algorithms. For the first time, we obtain a Waring rank upper bound that is exponential in the border Waring rank and only *linear* in the degree. All previous results were known to be exponential in the degree. 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 in this way we de-border Kumar's complexity, and obtain a new formulation of border Waring rank, up to a factor of the degree. We phrase this new formulation as the orbit closure problem of the product-plus-power polynomial, and we successfully de-border this orbit closure. We fully implement the GCT approach against the power sum, and we generalize 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 points and representation theoretic branching rules, rather than explicit multilinear computations. Furthermore, we realize that the generalization of our converse of Kumar's theorem to square matrices gives a homogeneous formulation of Ben-Or and Cleve (SICOMP'92). This results ...
Innovation
Methods, ideas, or system contributions that make the work stand out.
Proves converse of Kumar's border Waring rank result
Deborders orbit closures with algebraic branching programs
Implements GCT approach for power sum symmetries
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