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By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper…
Let $\K$ be a field of characteristic zero and $\Kbar$ be an algebraic closure of $\K$. Consider a sequence of polynomials$G=(g\_1,\dots,g\_s)$ in $\K[X\_1,\dots,X\_n]$, a polynomial matrix $\F=[f\_{i,j}] \in \K[X\_1,\dots,X\_n]^{p \times…
This paper presents the following results on sets that are complete for NP. 1. If there is a problem in NP that requires exponential time at almost all lengths, then every many-one NP-complete set is complete under length-increasing…
By some new recursive algorithms, in this paper, we will give some improvements on Waring's problem.
Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the…
We propose a consistent polynomial-time method for the unseeded node matching problem for networks with smooth underlying structures. Despite widely conjectured by the research community that the structured graph matching problem to be…
We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the B\'ezout number of the system and linear in its bit size. Our…
The paper deals with the analytic complexity of solutions to bivariate holonomic hypergeometric systems of the Horn type. We obtain estimates on the analytic complexity of Puiseux polynomial solutions to the hypergeometric systems defined…
Complex structures are typical in machine learning. Tailoring learning algorithms for every structure requires an effort that may be saved by defining a generic learning procedure adaptive to any complex structure. In this paper, we propose…
This article considers Hamiltonian mechanical systems with potential functions admitting jump discontinuities. The focus is on accurate and efficient numerical approximations of their solutions, which will be defined via the laws of…
Reducing the conditions under which a given set satisfies the stipulations of the subset sum proposition to a set of linear relationships, the question of whether a set satisfies subset sum may be answered in a polynomial number of steps by…
Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
Polymorphic variants are a useful feature of the OCaml language whose current definition and implementation rely on kinding constraints to simulate a subtyping relation via unification. This yields an awkward formalization and results in a…
Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives. The mentioned representatives are…
We study the homology groups of the complement of a complexified real line arrangement with coefficients in complex rank-one local systems. Using Borel--Moore homology, we establish an algorithm computing their dimensions via the real…
We connect the mixing behaviour of random walks over a graph to the power of the local-consistency algorithm for the solution of the corresponding constraint satisfaction problem (CSP). We extend this connection to arbitrary CSPs and their…
An improved characteristic set algorithm for solving Boolean polynomial systems is proposed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain…
Archetypal analysis is an unsupervised learning method for exploratory data analysis. One major challenge that limits the applicability of archetypal analysis in practice is the inherent computational complexity of the existing algorithms.…
We discuss the advantages and limitations of cyclotomic fields to have fast polynomial arithmetic within homomorphic encryption, and show how these limitations can be overcome by replacing cyclotomic fields by a family that we refer to as…