The Hamiltonian Cycle polynomial, denoted as $HC_n$, is defined to be the sum of the weighted Hamiltonian Cycles in an $n$-vertex complete digraph, with vertices labeled $1$ to $n$ and edges weighted by formal variables $x_{i,j}$. The Permanent and $HC$, defined as the family $\{HC_n | \ n \geq 1\}$, ... more >>>

We introduce an approach to distinguishing isomorphism types of graphs based on vector spaces of polynomials that are set-wise invariant under permutations (“separating modules,” which are representations of the symmetric group), inspired by the Geometric Complexity Theory approach to separating complexity classes (Mulmuley & Sohoni, SIAM J. Comput., 2001). We ... more >>>

We study factoring algorithms for general sparse polynomials and sparse polynomials of bounded individual degree and prove the following results.
1. We give a deterministic polynomial-time algorithm which takes as input an $n$-variate $s$-sparse polynomial $f$ of bounded individual degree $d$ and outputs a list of circuits which contains ... more >>>