Journal of Computational Algebra最新文献

We determine the conjugacy class fusion from certain maximal subgroups of the Monster to the Monster, to justify the addition of these data to the Character Table Library in the computational algebra system GAP. The maximal subgroups in question are $\left({\mathrm{PSL}}_2\right(11)\times {\mathrm{PSL}}_2(11\left)\right):4$, ${11}^2:(5\times 2{\text{A}}_5)$, $7^2:{\text{SL}}_2\left(7\right)$, and ${\mathrm{PSL}}_2\left(19\right):2$. Our proofs are supported by reproducible calculations carried out using the Python package mmgroup, a computational construction of the Monster recently developed by Seysen.

A Howe curve is defined as the normalization of the fiber product over a projective line of two hyperelliptic curves. Howe curves are very useful to produce important classes of curves over fields of positive characteristic, e.g., maximal, superspecial, or supersingular ones. Determining their feasible equations explicitly is a basic problem, and it has been solved in the hyperelliptic case and in the non-hyperelliptic case with genus not greater than 4. In this paper, we construct an explicit plane sextic model for non-hyperelliptic Howe curves of genus 5. We also determine the number and type of singularities on our sextic model, and prove that the singularities are generically 4 double points. Our results together with Moriya-Kudo's recent ones imply that for each $s\in \{2,3,4,5\}$, there exists a non-hyperelliptic curve H of genus 5 with $\mathrm{Aut}\left(H\right)\supset V_4$ such that its associated plane sextic has s double points.

In algebraic geometry or number theory, enumerating or finding superspecial curves in positive characteristic p is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group. By executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for all primes p with $19\le p<500$, and in finding a single one for all primes p with $19\le p<7000$.

In the past several years, Howe curves have been studied actively in the field of algebraic curves over fields of positive characteristic. Here, a Howe curve is defined as the desingularization of the fiber product over a projective line of two hyperelliptic curves. In this paper, we construct an explicit plane sextic model for non-hyperelliptic Howe curves of genus five. We also determine singularities of our sextic model. Some possible applications such as finding curves over finite fields of special properties are also described.

The Suwa method for computing versal unfoldings of holomorphic singular foliations is considered from the point of view of computational complex analysis. Based on the theory of Grothendieck local duality on residues, an effective algorithm of computing a first order infinitesimal versal unfoldings of codimension one complex analytic singular foliations is obtained. As an application of our approach, we give an effective method for computing universal unfoldings of germs of meromorphic functions.

First constructed by Fomin and Zelevinski [13], cluster algebras have been studied from many different perspectives. One such perspective is the study of cluster tilted algebras. We focus on when C is a monomial tilted algebras and $\tilde{C}$ its associated cluster tilted algebra. We show the set of partial derivatives of the Keller potential form a minimal set of relations for $\tilde{C}$ and we show that if C is also Koszul, then there are overlap relations that can be used to determine if $\tilde{C}$ is Koszul. We use the tools of noncommutative Gröbner basis theory to prove these results.

We formulate the Root Extraction problem in finite Abelian p-groups and then extend it to generic finite Abelian groups. We provide algorithms to solve them. We also give the bounds on the number of group operations required for these algorithms. We observe that once a basis is computed and the discrete logarithm relative to the basis is solved, root extraction takes relatively fewer “bookkeeping” steps. Thus, we conclude that root extraction in finite Abelian groups is no harder than solving discrete logarithms and computing basis.

Given a complex vector space V of finite dimension, its Grassmannian variety parametrizes all subspaces of V of a given dimension. Similarly, if a finite group G acts on V, its invariant Grassmannian parametrizes all the G−invariant subspaces of V of a given dimension. Based on this fact, we develop an algorithm for finding equations of G−invariant projective varieties arising as an intersection of hypersurfaces of the same degree.

We apply the algorithm to find equations describing a polarized K3 surface with a faithful action of $T_{192}⋊{\mu }_2$ and some further symmetric K3 surfaces with a degree 8 polarization.

We prove that a kG-module has a semilinear tensor decomposition if and only if its endomorphism algebra has a pair of mutually centralizing, unital, G-invariant subalgebras that are not commutative and are isomorphic to complete matrix algebras over an extension field K of k. We give an algorithm that constructs a semilinear tensor decomposition for any module whose endomorphism algebra contains appropriate invariant subalgebras.

Let $M$ be the Monster group, which is the largest sporadic finite simple group, and has first been constructed in 1982 by Griess. In 1985 Conway has constructed a 196884-dimensional rational representation ρ of $M$ with matrix entries in $Z\left[\frac{1}{2}\right]$. We describe a new and very fast algorithm for performing the group operation in $M$.

For an odd integer $p>1$ let ${\rho }_p$ be the representation ρ with matrix entries taken modulo p. We use a generating set Γ of $M$, such that the operation of a generator in Γ on an element of ${\rho }_p$ can easily be computed.

We construct a triple $(v_1,v^{+},v^{-})$ of elements of the module ${\rho }_{15}$, such that an unknown $g\in M$ can be effectively computed as a word in Γ from the images $(v_{1}g,v^{+}g,v^{-}g)$.

Our new algorithm based on this idea multiplies two random elements of $M$ in less than 30 milliseconds on a standard PC with an Intel i7-8750H CPU at 4 GHz. This is more than 100000 times faster than estimated by Wilson in 2013.