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Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism…
Determinantal polynomials play a crucial role in semidefinite programming problems. Helton-Vinnikov proved that real zero (RZ) bivariate polynomials are determinantal. However, it leads to a challenging problem to compute such a…
Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of…
The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear…
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…
We give a deterministic, polynomial-time algorithm for approximately counting the number of {0,1}-solutions to any instance of the knapsack problem. On an instance of length n with total weight W and accuracy parameter eps, our algorithm…
In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems)…
We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the…
In this paper we analyse the dynamics of a family of rational operators coming from a fourth-order family of root-finding algorithms. We first show that it may be convenient to redefine the parameters to prevent redundancies and…
While efficient randomized algorithms for factorization of polynomials given by algebraic circuits have been known for decades, obtaining an even slightly non-trivial deterministic algorithm for this problem has remained an open question of…
We study symmetric arithmetic circuits and improve on lower bounds given by Dawar and Wilsenach (ArXiv 2020). Their result showed an exponential lower bound of the permanent computed by symmetric circuits. We extend this result to show a…
The determinant for complex matrices cannot be extended to quaternionic matrices. Instead, the Study determinant and the closely related $q$-determinant are widely used. We show that the Study determinant can be characterized as the unique…
An efficient randomized polynomial identity test for noncommutative polynomials given by noncommutative arithmetic circuits remains an open problem. The main bottleneck to applying known techniques is that a noncommutative circuit of size…
We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are…
We provide a list of new natural $\mathsf{VNP}$-intermediate polynomial families, based on basic (combinatorial) $\mathsf{NP}$-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in…
We present a deterministic algorithm, which, for any given 0< epsilon < 1 and an nxn real or complex matrix A=(a_{ij}) such that | a_{ij}-1| < 0.19 for all i, j computes the permanent of A within relative error epsilon in n^{O(ln n -ln…
We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one…
Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient…
We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…
Artificial and biological neural networks (ANNs and BNNs) can encode inputs in the form of combinations of individual neurons' activities. These combinatorial neural codes present a computational challenge for direct and efficient analysis…