Related papers: On the complexity of class group computations for …

Using only undergraduate-level methods, we classify all groups of order $p^4$, where $p$ is an odd prime.

We propose an explicit and practical algorithm for computing Galois conjugates and irreducible polynomials for special values of modular functions evaluated at CM points associated with imaginary quadratic orders. Our approach builds upon…

We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all…

In this paper, we calculate the upper bound on quantum space complexity of the quantum algorithms proposed by Biasse and Song (SODA'16) for solving class group computation and the principal ideal problem using the reductions to $S$-unit…

Several recent deep neural networks experiments leverage the generalist-specialist paradigm for classification. However, no formal study compared the performance of different clustering algorithms for class assignment. In this paper we…

Massive classification, a classification task defined over a vast number of classes (hundreds of thousands or even millions), has become an essential part of many real-world systems, such as face recognition. Existing methods, including the…

We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices. The Zariski closure was previously shown to be computable by Derksen, Jeandel, and Koiran, but the termination argument for their…

We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in $X$ and $Y$ are low with respect to their genera. The finite base fields $\FF_q$ are arbitrary, but their sizes…

We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.

A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…

In [BDyDF08] Belabas, Diaz y Diaz and Friedman show a way to determine, assuming the Generalized Riemann Hypothesis, a set of prime ideals that generate the class group of a number field. Their method is efficient because it produces a set…

We survey group-theoretic algorithms for finding (some or all) subgroups of a finite group and discuss the implementation of these algorithms in the computer algebra system GAP

For polynomials of degree two over finite fields, we present an improvement of Fitzgerald's characterization (Finite Fields Appl. 9(1):117-121, 2003). We then use this new characterization to obtain an explicit, complete, and simple…

A new fast algorithm for clustering and classification of large collections of text documents is introduced. The new algorithm employs the bipartite graph that realizes the word-document matrix of the collection. Namely, the modularity of…

A robust theoretical framework that can describe and predict the generalization ability of deep neural networks (DNNs) in general circumstances remains elusive. Classical attempts have produced complexity metrics that rely heavily on global…

This article presents a methodology that automatically derives a combinatorial specification for a permutation class C, given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is…

Shanks's infrastructure algorithm and Buchmann's algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we…

In this paper we define Gevrey polyanalytic classes of order N on the unit disk D and we obtain for these classes a characteristic expansion into N-analytic polynomials on suitable neighborhoods of D. As an application of our main theorem,…

This paper provides an algorithmic generalization of Dickson's method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial…

We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification Rosenthal compacta, to identify new complexity classes. We use the language of…