arXiv subjects

Abbas Maarefparvar

Publications and source records attributed to Abbas Maarefparvar.

9 recordsLinked to original sources

arXiv2025

Pólya-Ostrowski Group and Unit Index in Real Biquadratic Fields

The Pólya-Ostrowski group of a Galois number field $K$, is the subgroup $Po(K)$ of the ideal class group $Cl(K)$ of $K$ generated by the classes of all the strongly ambiguous ideals of $K$. The number field $K$ is called a Pólya field, whenever $Po(K)$ is trivial. In this paper, using some results of Bennett Setzer \cite{Bennett} and Zantema \cite{Zantema}, we give an explicit relation between the order of Pólya groups and the Hasse unit indices in real biquadratic fields. As an application, we refine Zantema's upper bound on the number of ramified primes in Pólya real biquadratic fields.

math.NT↗

arXiv2025

Number Fields With Large Pólya Groups

The Pólya group ${\rm Po}(K)$ of a number field $K$ is the subgroup of the ideal class group ${\rm Cl}(K)$ of $K$ generated by the classes of all the products of the prime ideals of $K$ with the same norm. Motivated by the classical "one class in each genus problem", we prove general finiteness theorems for the number fields $K$ with a fixed Pólya index $\left[{\rm Cl}(K):{\rm Po}(K)\right]$ in the families of Galois number fields, solvable CM-fields, and real quadratic fields of extended R-D type. We also give classification results for specific families. Most notably, we classify, unconditionally, all imaginary bi-quadratic and imaginary tri-quadratic fields with the Pólya index one. Furthermore, we classify all real quadratic fields of extended R-D type (with possibly only one more field) with the Pólya index one. Also, under GRH, we give the complete list of 161 imaginary quadratic fields with the Pólya index two. Finally, as a byproduct of our results, we extend, from narrow R-D types to the extended R-D types, Dohmae's classification of real quadratic fields of narrow R-D type whose narrow genus numbers equal their narrow class numbers.

math.NT↗

arXiv2025

A Digital signature scheme based on Module-LWE and Module-SIS

In this paper, we present an improved version of the digital signature scheme proposed by Sharafi and Daghigh based on Module-LWE and Module-SIS problems. Our proposed signature scheme has a notably higher security level and smaller decoding failure probability, than the ones in the Sharaf-Daghigh scheme, at the expense of enlarging the module of the underlying basic ring.

cs.CR↗

arXiv2024

On Pólya groups of some non-Galois number fields

We prove two conjectures proposed by Chabert and Halberstadt concerning Pólya groups of $S_4$-fields and $D_4$-fields. More generally, the latter will be proved for $D_n$-fields with $n \geq 4$ an even integer. Further, generalizing a result of Zantema, we also prove that the pre-Pólya group of a non-Galois field of a prime degree, e.g. an $A_5$-field, coincides with its Pólya group.

math.NT↗

arXiv2023

The $S$-relative Pólya groups and $S$-Ostrowski quotients of number fields

Let $K/F$ be a finite extension of number fields and $S$ be a finite set of primes of $F$, including all the archimedean ones. In this paper, using some results of González-Avilés \cite{Aviles}, we generalize the notions of the relative Pólya group $\Po(K/F)$ \cite{ChabertI,MR2} and the Ostrowski quotient $\Ost(K/F)$ \cite{SRM} to their $S$-versions. Using this approach, we obtain generalizations of some well-known results on the $S$-capitulation map, including an $S$-version of Hilbert's theorem 94.

math.NT↗

arXiv2023

The Ostrowski quotient of an elliptic curve

For $K/F$ a finite Galois extension of number fields, the relative Pólya group $\Po(K/F)$ is the subgroup of the ideal class group of $K$ generated by all the strongly ambiguous ideal classes in $K/F$. The notion of Ostrowski quotient $\Ost(K/F)$, as the cokernel of the capitulation map into $\Po(K/F)$, has been recently introduced in \cite{SRM}. In this paper, using some results of González-Avilés \cite{Aviles}, we find a new approach to define $\Po(K/F)$ and $\Ost(K/F)$ which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For $E$ an elliptic curve defined over $F$, we define the Ostrowski quotient $\Ost(E,K/F)$ and the coarse Ostrowski quotient $\Ost_c(E,K/F)$ of $E$ relative to $K/F$, for which in the latter group we do not take into account primes of bad reduction. Our main result is a non-trivial structure theorem for the group $\Ost_c(E,K/F)$ and we analyze this theorem, in some detail, for the class of curves $E$ over quadratic extensions $K/F$.

math.NT↗

arXiv2022

Ostrowski quotients for finite extensions of number fields

For $L/K$ a finite Galois extension of number fields, the relative Pólya group $\Po(L/K)$ coincides with the group of strongly ambiguous ideal classes in $L/K$. In this paper, using a well known exact sequence related to $\Po(L/K)$, in the works of Brumer-Rosen and Zantema, we find short proofs for some classical results in the literatur. Then we define the ``Ostrowski quotient'' $\Ost(L/K)$ as the cokernel of the capitulation map into $\Po(L/K)$, and generalize some known results for $\Po(L/\mathbb{Q})$ to $\Ost(L/K)$.

math.NT↗

arXiv2020

Robustness Analysis of the Data-Selective Volterra NLMS Algorithm

Recently, the data-selective adaptive Volterra filters have been proposed; however, up to now, there are not any theoretical analyses on its behavior rather than numerical simulations. Therefore, in this paper, we analyze the robustness (in the sense of l2-stability) of the data-selective Volterra normalized least-mean-square (DS-VNLMS) algorithm. First, we study the local robustness of this algorithm at any iteration, then we propose a global bound for the error/discrepancy in the coefficient vector. Also, we demonstrate that the DS-VNLMS algorithm improves the parameter estimation for the majority of the iterations that an update is implemented. Moreover, we prove that if the noise bound is known, we can set the DS-VNLMS so that it never degrades the estimate. The simulation results corroborate the validity of the executed analysis and demonstrate that the DS-VNLMS algorithm is robust against noise, no matter how its parameters are adopted.

cs.LG↗

arXiv2019

Relative Polya Group and Polya Dihedral Extensions of Q

We define the relative Polya group for a finite extension of number fields and prove triviality of the relative Polya group for the Hilbert class field. Then we generalize our previous results on Polya S3-extensions of Q to some dihedral extensions of Q.

math.NT↗