Rongyu Xu | Searcharxiv

arXiv2023

On the Chow-rank of the permanent

We derive Glynn's formula from Ryser's formula for the permanent. We further establish via an orbital argument that Glynn's formula yields an optimal row-homogeneous Chow-decomposition of the permanent. We introduce a method for upper-bounding the Chow-rank of polynomials. Our method is based upon the Fundamental Theorem of Symmetric Polynomials. Finally we derive a parametric description of rank revealing row-homogeneous Chow-decompositions of the permanent. Our result provides an alternative approach to the parametrization first obtained by Glynn.

cs.CC↗

arXiv2022

On Partial Differential Encodings of Boolean Functions

We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to specify partial differential encodings of Boolean functions. We also introduce variants of partial differential encodings called partial differential programs. We show that such programs optimally describe important families of polynomials including determinants and permanents. Partial differential programs also enables to quantitively contrast these two families of polynomials. Finally we derive from polynomial constructions inspired by partial differential programs which exhibit an unconditional exponential separation between high order hypergraph isomorhism instances and their sub-isomorphism counterparts.

cs.CC↗