Subset Sum in $O(n^{11}\log(n))$

Abstract

This paper describes an algorithm (thus far referred to as the "Dragonfly Algorithm") by which the subset-sum problem can be solved in $O(n^{11}\log(n))$ time complexity. The paper will first detail the generalized "product-derivative" method (and the more efficient version of this method which will be used in the algorithm) by which a pair of monic polynomials can be used to generate a system of unique monic polynomials for which each polynomial in the system will share with every other a set of roots equivalent to the intersection of the roots of the original pair; this method will then be applied on a pair of polynomials one of which, $\phi(x)$, exhibiting known roots based on the instance of the subset-sum problem and the other of which, $t(x)$, containing unknown placeholder coefficients and representing an unknown subset of the linear factors of $\phi(x)$.