Related papers: Computing Galois groups of Ehrhart polynomials in …
A generalisation of the odd Bernoulli polynomials related to the quantum Euler top is introduced and investigated. This is applied to compute the coefficients of the spectral polynomials for the classical Lam\'e operator.
For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the…
In 1992, Avner Ash and Mark McConnell presented computational evidence of a connection between three-dimensional Galois representations and certain arithmetic cohomology classes. For some examples they were unable to determine the attached…
For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the…
A lot of work has gone into computing images of Galois representations coming from elliptic curves. This article presents an algorithm to determine the image of the mod-$3$ Galois representation associated to a principally polarized abelian…
We obtain a new classification of the finite metacyclic group in terms of group invariants. We present an algorithm to compute these invariants, and hence to decide if two given finite metacyclic groups are isomorphic, and another algorithm…
This is the first one of a series of papers on association of orientations, lattice polytopes, and abelian group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative…
Euler systems are certain compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. We briefly survey the known Euler systems, and recall a standard conjecture of Perrin-Riou…
The polynomial algebra is a deformed SU(2) algebra. Here, we use polynomial algebra as a method to solve a series of deformed oscillators. Meanwhile, we find a series of physics systems corresponding with polynomial algebra with different…
We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending…
Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of…
We present an algorithm for computing the set of torsion points satisfying a given system of multivariate polynomial equations. Its complexity is quasilinear in the logarithm of the degree of the input equations and exponential in their…
We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit…
Let $\mathcal{E}$ be an elliptic curve over a field $K$ and $\ell$ a prime. There exists an elliptic curve $\mathcal{E}^*$ related to $\mathcal{E}$ by anisogeny (rational map that is also a group homomorphisms) of degree $\ell$ if and only…
Gr{\"o}bner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases,…
For real number $\alpha,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by \begin{align*} L_n^{(\alpha)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+\alpha}{n-j}\frac{(-x)^j}{j!}. \end{align*}These orthogonal…
We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a…
We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $\ell^n$ where $\ell$ is the lacunary size of the input polynomial and $n$…
Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible…
We compute the virtual Poincar\'e polynomials of the configuration space of $n$ ordered points on an elliptic curve with sum 0.