Related papers: Computing the fixing group of a rational function

Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality…

The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a…

For a given l-adic sheaf F on a commutative algebraic group over a finite field k and an integer r we define the r-th local norm L-function of F at a point t in G(k) and prove its rationality. This function gives information on the sum of…

Field Arithmetic studies the interplay between arithmetical properties of fields and their absolute Galois groups. Here we studies fields satisfying local global principles for rational points of varieties and profinite groups satisfying…

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…

We classify all rational maps $H \in K(x)^n$ for which ${\rm trdeg}_K K(tH_1,tH_2,\ldots,tH_n) \le 2$, where $K$ is any field and $t$ is another indeterminate. Furthermore, we classify all such maps for which additionally $JH \cdot H = {\rm…

We describe algorithms to compute fixed fields, splitting fields and towers of radical extensions without using polynomial factorisation in towers or constructing any field containing the splitting field, instead extending Galois group…

We propose a polynomial time $f$-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$) for computing an isomorphism (if there is any) of a finite dimensional…

We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…

Let $k$ be an algebraic closed field of characteristic zero. Let $K$ be the rational function field $K=k(t)$. Let $\phi$ be a non isotrivial rational function in $K(z)$. We prove a bound for the cardinality of the set of $K$--rational…

In this note, we state a theorem of compution of the unipotent radical of the Galois group of an object $U$ of a tannakian category defined over a field of positive characteristic, extension of the unit object by a semi-simple one. We then…

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…

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if $K$ is a number field and $L/K$ is an infinite Galois…

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$-automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's…

We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over $\mathbb…

Let $F$ be a finite field of odd cardinality $q$, $A=F[T]$ the polynomial ring over $F$, $k=F(T)$ the rational function field over $F$ and $\mathcal{H}$ the set of square-free monic polynomials in $A$ of degree odd. If $D\in\mathcal{H}$, we…

We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$…

Let $\C$ be a sequence of multisets of subspaces of a vector space $\F_q^k$. We describe a practical algorithm which computes a canonical form and the stabilizer of $\C$ under the group action of the general semilinear group. It allows us…

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the…

This paper studies the number of monic integer polynomials $f$ of height at most $H$ whose Galois group, endowed with the action on the roots, is isomorphic to a prescribed permutation group $(G,\Omega)$. New upper bounds are obtained for…