Related papers: On the generalised Ritt problem as a computational…
We use the generalized concurrence approach to investigate the general multipartite separability problem. By extending the preconcurrence matrix formalism to arbitrary multipartite systems, we show that the separability problem can be…
We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials…
We believe we have made progress in the age-old problem of divisibility rules for integers. Universal divisibility rule is introduced for any divisor in any base number system. The divisibility criterion is written down explicitly as a…
Some techniques for the use of bitwise operations are described in the article. As an example, an open problem of isomorphism-free generations of combinatorial objects is discussed. An equivalence relation on the set of square binary…
Current algorithms for large-scale industrial optimization problems typically face a trade-off: they either require exponential time to reach optimal solutions, or employ problem-specific heuristics. To overcome these limitations, we…
Checking two probabilistic automata for equivalence has been shown to be a key problem for efficiently establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test…
We describe an algorithm which finds binomials in a given ideal $I\subset\mathbb{Q}[x_1,\dots,x_n]$ and in particular decides whether binomials exist in $I$ at all. Binomials in polynomial ideals can be well hidden. For example, the lowest…
An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a…
Given a convex optimization problem and its dual, there are many possible first-order algorithms. In this paper, we show the equivalence between mirror descent algorithms and algorithms generalizing the conditional gradient method. This is…
For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a…
For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is presented, provided it is zero-dimensional (even if $I$ is not). Moreover we propose an…
Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…
Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be…
The main point of this paper is to present a class of equations over integers that one can check if they have a solution by checking a set of inequalities. The prototype of such equations is the equations appearing in the well-known…
Differential algebra approaches to structural identifiability analysis of a dynamic system model in many instances heavily depend upon Ritt's pseudodivision at an early step in analysis. The pseudodivision algorithm is used to find the…
We study the quadratic residue problem known as an NP complete problem by way of the prime number and show that a nondeterministic polynomial process does not belong to the class P because of a random distribution of solutions for the…
Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a…
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…
In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time…
The Profinite Isomorphism Problem for a class of groups \mathcal{C} asks for an algorithm that decides for any two groups in \mathcal{C} whether they have isomorphic profinite completions. We present the positive solution to this problem…