Related papers: Application of Weierstrass units to relative power…

Let $M\subset K$ be number fields. We consider the relation of relative power integral bases of $K$ over $M$ with absolute power integral bases of $K$ over $Q$. We show how generators of absolute power integral bases can be calculated from…

Investigations of monogenity and power integral bases were recently extended from the absolute case (over Q) to the relative case (over algebraic number fields). Formerly, in the relative case we only succeeded to calculate generators of…

Let $M=Q(i\sqrt{d})$ be any imaginary quadratic field with a positive square-free $d$. Consider the polynomial \[ f(x)=x^3-ax^2-(a+3)x-1, \] with a parameter $a\in Z$. Let $K=M(\alpha)$, where $\alpha$ is a root of $f$. This is an infinite…

We consider infinite parametric families of octic fields, that are quartic extensions of quadratic fields. We describe all relative power integral bases of the octic fields over the quadratic subfields.

In our recent paper we gave an efficient algorithm to calculate "small" solutions of relative Thue equations (where "small" means an upper bound of type $10^{500}$ for the sizes of solutions). Here we apply this algorithm to calculating…

Some time ago we extended our monogenity investigations and calculations of generators of power integral bases to the relative case. Up to now we considered (usually totally real) extensions of complex quartic fields. In the present paper…

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a nontrivial integral ideal $\mathfrak{m}$ of $K$, let $K_\mathfrak{m}$ be the ray class field modulo $\mathfrak{m}$. By using…

Let $M$ be an imaginary quadratic field with the ring of integers $\mathbb{Z}_{M}$ and let $\xi$ be a root of polynomial $$f\left( x\right) =x^{4}-2cx^{3}+2x^{2}+2cx+1,$$ where $c\in\mathbb{Z}_{M},$ $c\notin\left\{ 0,\pm2\right\}$. We…

We develop a criterion for a normal basis, and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$.…

A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle.It was suggested that such an equation may find application in gravity…

Let $K$ be a finite extension of $\Q_p$, let $L/K$ be a finite abelian Galois extension of odd degree and let $\bo_L$ be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional $\bo_L$-ideal with square equal to the…

We study the properties of special values of the modular functions obtained from Weierstrass P-function at imaginary quadratic points.

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_\mathfrak{n}$ be the ray class field of $K$ modulo $\mathfrak{n}=N\mathcal{O}_K$. By using the…

A closed form expression for the three-electron Hylleraas integral involving the inverse quadratic power of one inter-particle coordinate is obtained, and recursion relations are derived for positive powers of other coordinates. This result…

We describe a systematic approach to the construction of loop-integrand bases at arbitrary loop-order, sufficient for the representation of general quantum field theories. We provide a graph-theoretic definition of `power-counting' for…

This paper studies the integration problem in differential fields that may involve quantities reminiscent of the Weierstrass $\wp$ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion…

We describe a nonstandard version of the quantum plane, the one in the basis of divided powers at an even root of unity $q=e^{i\pi/p}$. It can be regarded as an extension of the "nearly commutative" algebra $C[X,Y]$ with $X Y =(-1)^p Y X$…

We rephrase the Wootters-Fields construction [Ann. Phys., {\bf 191}, 363 (1989)] of a full set of mutually unbiased bases in a complex vector space of dimensions $N=p^r$, where $p$ is an odd prime, in terms of the character vectors of the…

We first normalize the derivative Weierstrass $\wp'$-function appearing in Weierstrass equations which give rise to analytic parametrizations of elliptic curves by the Dedekind $\eta$-function. And, by making use of this normalization of…

The generalized Weierstrass representation for surfaces in $\Bbb{R}^{3}$ is used to study quantum effects for strings governed by Polyakov-Nambu-Goto action. Correlators of primary fields are calculated exactly in one-loop approximation for…