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Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so called generalized toric codes. This extension consists on evaluating elements of an…
In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…
Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path) integrals by gauge theory or non-linear…
Known mappings that encode fermionic modes into a bosonic qubit system are non-local transformations. In this paper we establish that this must necessarily be the case, if the locality graph is complex enough (for example for regular 2$d$…
Employing isomorphisms between their ambient rings, we propose new definitions of equivalence and isometry for skew polycyclic codes that will lead to tighter classifications than existing ones. This reduces the number of previously known…
The purpose of this article is to study polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}, \,t \geq 1$, and their associated torsion codes. It is shown that if $\phi$ is a surjective ring homomorphism from a…
We present selected examples demonstrating an alternative approach to contour deformation for numerically computing loop integrals in the Minkowski regime. This method focuses on identifying singular hypersurfaces (varieties of the…
The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant…
We prove that generating subspaces of matrix rings over finite fields are counted by polynomials. We use this result to define and study two-variable versions of polynomials counting isomorphism classes of absolutely irreducible…
In this paper, we study the codes over the matrix ring over $\mathbb{Z}_4$, which is perhaps the first time the ring structure $M_2(\mathbb{Z}_4)$ is considered as a code alphabet. This ring is isomorphic to…
We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…
In this paper, by using the theory of circulant matrices we study some matrices over finite fields which involve the quadratic character and trinomial coefficients.
This paper studies hamiltonization of nonholonomic systems using geometric tools. By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic…
It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification…
Non-Hermitian systems exploiting the synergy between gain and loss have recently become the focus of interest to discover novel physical phenomena. The spatial symmetry breaking in such systems allows tailoring the wave propagation at will.…
For a complex polynomial in two variables we study the morphism induced in homology by the embedding of an irregular fiber in a regular neighborhood of it. We give necessary and sufficient conditions for this morphism to be injective,…
We introduce the notion of "quasi-symmetric" polynomials, which is a generalization of the notion of symmetry, and is particularly suited to the setting of polynomial rings over finite fields. The properties of this new class of functions…
This paper presents fundamental algorithms for the computational theory of quadratic forms over number fields. In the first part of the paper, we present algorithms for checking if a given non-degenerate quadratic form over a fixed number…
We investigate the rings of semi-invariants for tame string algebras A(n) of non-polynomial growth. We are interested in dimension vectors of band modules. We use geometric technique related to the description of coordinate rings on…
In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic…