Journal of Computational Algebra最新文献

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.

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.

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.

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.

We give generating sets of the Jacobson radical of the hyperalgebra of the r-th Frobenius kernel of the algebraic group ${\mathrm{SL}}_2$ over an algebraically closed field of characteristic $p>0$. This result generalizes earlier work by Wong for $r=1$ and odd p.

We describe a new algorithm for finding a canonical image of an object under the action of a finite permutation group. This algorithm builds on previous work using Graph Backtracking [9], which extends Jeffrey Leon's Partition Backtrack framework [14], [15]. Our methods generalise both Nauty [17] and Steve Linton's Minimal image algorithm [16].