Related papers: Algorithmic problems for differential polynomial a…
The main result of this paper is the decidability of the membership problem for $2\times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2\times 2$ integer matrices $M_1,\dots,M_n$…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial…
We consider the polynomial Ideal Membership Problem (IMP) for ideals encoding combinatorial problems that are instances of CSPs over a finite language. In this paper, the input polynomial $f$ has degree at most $d=O(1)$ (we call this…
We make a connection between the subgroup membership and identity problems for matrix groups and extended finite automata. We provide an alternative proof for the decidability of the subgroup membership problem for $ 2 \times 2 $ integer…
This work presents a sample constructions of two algebras both with the ideal of relations defined by a finite Gr\"obner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the…
We show that for any class of Boolean algebras with an associative operator, if it contains the complex algebra of (P(N), U), its equational theory is undecidable. Equivalently, any associative normal modal logic valid over the frame (P(N),…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with…
We generalize the differential dimension polynomial from prime differential ideals to characterizable differential ideals. Its computation is algorithmic, its degree and leading coefficient remain differential birational invariants, and it…
We show that the rational subset membership problem in $G$ can be reduced to the submonoid membership problem in $G{\times}H$ where $H$ is virtually Abelian. We use this to show that there is no algorithm reducing submonoid membership to a…
The ideal membership problem asks whether an element in the ring belongs to the given ideal. In this paper, we propose a function that reflecting the complexity of the ideal membership problem in the ring of Laurent polynomials with integer…
We prove that the scalar and $2\times 2$ matrix differential operators which preserve the simplest scalar and vector-valued polynomial modules in two variables have a fundamental Lie algebraic structure. Our approach is based on a general…
The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…
We show that 2D periodic operators with local and perpendicular defects form an algebra. We provide an algorithm of finding spectrum for such operators. While the continuous spectral components can be computed by simple algebraic operations…
We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group…
We find a relation between the vanishing of a globally defined residue current on $\P^n$ and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max…
The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain…
Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the…