Related papers: Linearizing the Word Problem in (some) Free Fields
Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…
Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the…
We give a ranker-based description using finite-index congruences for the variety $\boldsymbol{\mathrm{DAb}}$ of finite monoids whose regular $\mathcal{D}$-classes form Abelian groups. This combinatorial description yields a normal form for…
An open question in the theory of word-representable graphs for the past decade has been whether the line graph of a non-word-representable graph is always non-word-representable. By formulating an appropriate optimization problem for the…
By a result of Helton and McCullough, open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D_L of a linear matrix inequality (LMI) L(X)>0. This paper gives a precise algebraic certificate for a polynomial…
In this paper we discourse basises of representable algebras. This question lead to arithmetic problems. We prove algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive…
Schrodinger equation for a charged particle interacting with the plane wave electromagnetic field is solved exactly. The exact analytic solution and the perturbative solution up to second order are compared.
To cater to the needs of (Zero Knowledge) proofs for (mathematical) proofs, we describe a method to transform formal sentences in 2x2-matrices over multivariate polynomials with integer coefficients, such that usual proof-steps like…
In this paper we present an application of a simple technique of local recompression, previously developed by the author in the context of compressed membership problems and compressed pattern matching, to word equations. The technique is…
The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex basic non-commutative semi-algebraic set. The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and…
We design new deterministic and randomized algorithms for computational problems in free solvable groups. In particular, we prove that the word problem and the power problem can be solved in quasi-linear time and the conjugacy problem can…
Free field representation for the classical limit of quantum affine algebra is constructed by simple deformation of the known expressions from WZW theory.
We study membership problems in HNN extensions of free groups and then apply these results to solve the word problem in certain families of one-relator inverse monoids. In more detail, we consider HNN extensions where the defining…
We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.
Let $ X $ be an $ m \times n $ matrix of distinct indeterminates over a field $ K $, where $ m \le n $. Set the polynomial ring $ K[X] := K[X_{ij} : 1 \le i \le m, 1 \le j \le n] $. Let $ 1 \le k < l \le n $ be such that $ l - k + 1 \ge m…
The offset method for solving word analogies has become a standard evaluation tool for vector-space semantic models: it is considered desirable for a space to represent semantic relations as consistent vector offsets. We show that the…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
For linear operators which factor with suitable assumptions concerning commutativity of the factors, we introduce several notions of a decomposition. When any of these hold then questions of null space and range are subordinated to the same…
We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…
We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular…