Rings Of Quadratic Integers Research Articles

Idempotent factorization on some matrices over quadratic integer rings

In 2020, Cossu and Zanardo raised a problem on the idempotent factorization of singular matrices in the form ( p z z ¯ ‖ z ‖ / p ) , where p is a prime integer which is irreducible but not prime in the ring of integers O K , with K a real quadratic number field, and z ∈ O K is such that 〈 p , z 〉 is a non-principal ideal. In this paper, we provide some classes of matrices that factor in a prescribed way and some other classes that do not. We further show that there are matrices in the above form that cannot be written as a product of two idempotent matrices.

On Farrell–Tate cohomology of GL3 over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on mod ℓ Farrell–Tate cohomology for arithmetic groups. We deal with ℓ-rank 1 cases different from PSL2. The conjugacy classification of cyclic subgroups of order ℓ is reduced to the classification of modules of Cℓ-group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell–Tate cohomology computations, we discuss in detail the homological torsion in PGL3 over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.

On lattice-based algebraic feedback shift registers synthesis for multisequences

In this paper we show that algebraic feedback shift registers synthesis problems over residue class rings, some ramified extensions and some quadratic integer rings for multisequences are reduced to the successive minima problem in lattice theory. Therefore they can be solved by polynomial-time algorithms since the number of multiple sequences is fixed.

Small-span Hermitian matrices over quadratic integer rings

A totally-real polynomial in Z [ x ] \mathbb {Z}[x] with zeros α 1 ⩽ α 2 ⩽ ⋯ ⩽ α d \alpha _1 \leqslant \alpha _2 \leqslant \dots \leqslant \alpha _d has span α d − α 1 \alpha _d - \alpha _1 . Building on the classification of all characteristic polynomials of integer symmetric matrices having small span (span less than 4 4 ), we obtain a classification of small-span polynomials that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. Taking quadratic integer rings as our base, we obtain as characteristic polynomials some low-degree small-span polynomials that are not the characteristic (or minimal) polynomial of any integer symmetric matrix.

Cyclotomic matrices over real quadratic integer rings

We classify all cyclotomic matrices over real quadratic integer rings and we show that this classification is the same as classifying cyclotomic matrices over the compositum all real quadratic integer rings, R. Moreover, we enumerate a related class of symmetric R-matrices; those R-matrices whose eigenvalues are contained inside the interval [−2,2] but whose characteristic polynomials are not in Z[x].

Lower bounds for decision problems in imaginary, norm-Euclidean quadratic integer rings

We prove lower bounds for the complexity of deciding several relations in imaginary, norm-Euclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer ring, a known binary-like gcd algorithm has complexity that is quadratic in our lower bound.

Conference Matrices and Unimodular Lattices

Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by nonprincipal ideals yields simple constructions of further lattices including the Leech lattice.

Good reduction of elliptic curves over imaginary quadratic fields

We prove that the j-invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.

A Thue Equation with Quadratic Integers as Variables

In an earlier paper we determined all the solutions in Z of a cubic Thue equation with coefficients in a quadratic number field. It is now shown that the method used there can be used to solve the more general problem of determining all the solutions of the Thue equation in a ring of quadratic integers.

A Thue equation with quadratic integers as variables

In an earlier paper we determined all the solutions in Z \mathbb {Z} of a cubic Thue equation with coefficients in a quadratic number field. It is now shown that the method used there can be used to solve the more general problem of determining all the solutions of the Thue equation in a ring of quadratic integers.

Homomorphisms of two-dimensional linear groups over Laurent polynomial rings and Gaussian domains

Let GL2(R) be the general linear group of 2 × 2 invertible matrices in M2(R) over a commutative ring R with 1 and SL2(R) be the special linear group consisting of 2 × 2 matrices over R with determinant 1. In this paper we determine the homomorphisms from GL2 and SL2, as well as their projective groups, over Laurent polynomial rings to those groups over Gaussian domains, i.e. unique factorization domains (cf. Theorems 1, 2, 3 below). We also consider more generally the homomorphisms of non-projective groups over commutative rings containing a field which are generated by their units (cf. Theorems 4 and 5). So far the homomorphisms of two-dimensional linear groups over commutative rings have only been studied in some specific cases. Landin and Reiner[7] obtained the automorphisms of GL2(R), where R is a Euclidean domain generated by its units. When R is a type of generalized Euclidean domain with a degree function and with units of R and 0 forming a field, Cohn[3] described the automorphisms of GL2(R). Later, Cohn[4] applied his methods to the case of certain rings of quadratic integers. Dull[6] has considered the automorphisms of GL2(R) and SL2(R), along with their projective groups, provided that R is a GE-ring and 2 is a unit in R. McDonald [9] examined the automorphisms of GL2(R) when R has a large unit group. The most recent work of which we are aware is that of Li and Ren[8] where the automorphisms of E2(R) and GE2(R) were determined for any commutative ring R in which 2, 3 and 5 are units.

Convolutions over residue classes of quadratic integers

A Fourier-like transform is defined over a ring of quadratic integers modulo a prime number <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> in the quadratic field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(\sqrt{m})</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is a square-free integer. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> is a Fermat prime, one can utilize the fast Fourier transform (FFT) algorithm over the resulting finite fields to yield fast convolutions of quadratic integer sequences in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(\sqrt{m})</tex> . The theory is also extended to a direct sum of such finite fields. From these results, it is shown that Fourier-like transforms can also be defined over the quadratic integers in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R( \sqrt{m})</tex> modulo a nonprime Fermat number.