Factorization in Quantum Planes

We give an algorithm for factoring quadratic polynomials over any UFD, Z in particular. We prove the correctness of this algorithm and give examples over Z and Z[i].

The goal of this paper is to introduce some rings that play the role of the jet spaces of the quantum plane and unlike the quantum plane itself possess interesting nontrivial prime ideals. We will prove some results (theorems 1-4) about the prime spectrum of these rings.

Let R be a Noetherian domain and let ({\sigma}, {\delta}) be a quasi-derivation of R such that {\sigma} is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F = t^2 - v is normal in the Ore extension R[t; {\sigma}, {\delta}] where v {\epsilon} R. We show F is prime in R[t; {\sigma}, {\delta}] if and only if F is irreducible in Q[t; {\sigma}, {\delta}] if and only if there does not exist w {\epsilon} Q such that v = {\sigma} (...

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing some sufficiency conditions on their coefficients along with some conditions on the prime factorization of constant term or the leading coefficient of the underlying polynomials in conjunction with the information about their root location.

The quantum plane is the non-commutative polynomial algebra in variables $x$ and $y$ with $xy=qyx$. In this paper, we study the module variety of $n$-dimensional modules over the quantum plane, and provide an explicit description of its irreducible components and their dimensions. We also describe the irreducible components and their dimensions of the GIT quotient of the module variety with respect to the conjugation action of ${\rm GL}_n$.

Let $p$ be an odd prime, $k,\ell$ be positive integers, $q=p^k, Q=p^{\ell}$. In this paper we characterise planar functions of the form $f_{\underline{c}}(X)=c_0X^{qQ+q}+c_1X^{qQ+1}+c_2X^{Q+q}+c_3X^{Q+1}$ over $\mathbb{F}_{q^2}$ for any $\underline{c}=(c_0,c_1,c_2,c_3) \in \mathbb{F}_{q^2}^4$ in terms of linear equivalence.

We develop a method of reducing the size of quantum minors in the algebra of n x n quantum matrices. The method is used to show that quantum determinantal factor rings of n x n quantum matrices over the complex numbers are maximal orders, when the parameter q is transcendental over the rational numbers.

A sequence of functions {f_n(q)}_{n=1}^{\infty} satisfies the functional equation for multiplication of quantum integers if f_{mn}(q) = f_m(q)f_n(q^m) for all positive integers m and n. This paper describes the structure of all sequences of rational functions with rational coefficients that satisfy this functional equation.

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is reducible in $Z[[x]]$ if and only if it is reducible in $Z_p[x]$, the ring of polynomials over the $p$-adic integers.

For the quantum integer $[n]_q = 1+q+...+q^{n-1}$ there is a natural polynomial multiplication $*_q$ such that $[m]_q *_q [n]_q = [mn]_q$. This multiplication leads to the functional equation $f_{mn}(q) = f_m(q)f_n(q^m),$ defined on a given sequence $\mathcal(F)=\{f_n(q)\}_{n=1}^{\infty}$ of polynomials. This paper contains various results concerning the classification and construction of polynomial sequences that satisfy the functional equation, as well as a list of open pro...