Related papers: A note on generators of number fields

We give lower bounds on the number of effective divisors of degree $\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and…

Let $k \subset {\mathbb C}$ be a number field and ${\mathcal E}$ be an elliptic curve defined over $k(t)$, the rational function field of the projective line ${\mathbb P}^1_k$, is isomorphic to the generic fiber of an elliptic surface…

We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the…

We construct explicit pseudorandom generators that fool $n$-variate polynomials of degree at most $d$ over a finite field $\mathbb{F}_q$. The seed length of our generators is $O(d \log n + \log q)$, over fields of size exponential in $d$…

Ribet has proven remarkable results about non-optimal levels of residually reducible Galois representations. We focus on a non-optimal level $N$ that is the product of two distinct primes and where the Galois deformation ring is not…

Let $\Gamma$ be a finitely generated subgroup of the multiplicative group $\G_m^2(\bar{Q})$. Let $p(X,Y),q(X,Y)\in\bat{Q}$ be two coprime polynomials not both vanishing at $(0,0)$; let $\epsilon>0$. We prove that, for all $(u,v)\in\Gamma$…

For a group $G$ and a finite set $A$, denote by $\text{CA}(G;A)$ the monoid of all cellular automata over $A^G$ and by $\text{ICA}(G;A)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of…

Let $U/L$ be a finite abelian extension of number fields. We first construct a universal primitive generator of $U$ over $L$ whose relative trace to any intermediate field $F$ becomes a generator of $F$ over $L$, too. We also develop a…

Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with…

For a set $E\subset \mathbb F_q^d$, we define the $k$-resultant magnitude set as $ \Delta_k(E) =\{\|\textbf{x}_1 + \dots + \textbf{x}_k\|\in \mathbb F_q: \textbf{x}_1, \dots, \textbf{x}_k \in E\},$ where $\|\textbf{v}\|=v_1^2+\cdots+ v_d^2$…

Let $K/\mathbb{Q}$ be an algebraic extension of fields, and let $\alpha \not= 0$ be contained in an algebraic closure of $K$. If $\alpha$ can be approximated by roots of numbers in $K^{\times}$ with respect to the Weil height, we prove that…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

We say that collection of $n$-qudit gates is universal if there exists $N_0\geq n$ such that for every $N\geq N_0$ every $N$-qudit unitary operation can be approximated with arbitrary precision by a circuit built from gates of the…

Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that…

In this paper we construct Galois towers with good asymptotic properties over any non-prime finite field $\mathbb F_{\ell}$; i.e., we construct sequences of function fields $\mathcal{N}=(N_1 \subset N_2 \subset \cdots)$ over $\mathbb…

Explicit bounds are given on the norms of prime ideals generating arbitrary subgroups of ray class groups of number fields, assuming the Extended Riemann Hypothesis. These are the first explicit bounds for this problem, and are…

A conjecture of Malle predicts the quantity of number fields with bounded discriminant of given Galois group. We present a lower bound matching this in the case of quartic fields with Galois group $A_4$.

The rank of a point-line geometry G is usually defined as the generating rank of G, namely the minimal cardinality of a generating set. However, when the subspace lattice of G satisfies the Exchange Property we can also try a different…

In this paper, assuming the weak Schanuel Conjecture (WSC), we prove that for any collection of pairwise non-arithmetically equivalent totally real number fields, the residues at $s=1$ of their Dedekind zeta functions form a linearly…

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an…