Solving conics over function fields

Mark van Hoeij; John Cremona

Displaying similar documents to “Solving conics over function fields”

Factoring polynomials over global fields

Karim Belabas, Mark van Hoeij, Jürgen Klüners, Allan Steel (2009)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.

Practical Aurifeuillian factorization

Bill Allombert, Karim Belabas (2008)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We describe a simple procedure to find Aurifeuillian factors of values of cyclotomic polynomials $Φ_{d} (a)$ for integers $a$ and $d > 0$ . Assuming a suitable Riemann Hypothesis, the algorithm runs in deterministic time $\tilde{O} (d^{2} L)$ , using $O (d L)$ space, where $L ≔ log (|a| + 1)$ .

Fast computation of class fields given their norm group

Loïc Grenié (2008)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let $K$ be a number field containing, for some prime $ℓ$ , the $ℓ$ -th roots of unity. Let $L$ be a Kummer extension of degree $ℓ$ of $K$ characterized by its modulus $𝔪$ and its norm group. Let $K_{𝔪}$ be the compositum of degree $ℓ$ extensions of $K$ of conductor dividing $𝔪$ . Using the vector-space structure of $Gal (K_{𝔪} / K)$ , we suggest a modification of the function of which brings the complexity of the computation of an equation of $L$ over $K$ from exponential to linear.

The fluctuations in the number of points on a family of curves over a finite field

Maosheng Xiong (2010)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let $l \geq 2$ be a positive integer, $𝔽_{q}$ a finite field of cardinality $q$ with $q \equiv 1 (mod l)$ . In this paper, inspired by [, , ] and using a slightly different method, we study the fluctuations in the number of $𝔽_{q}$ -points on the curve $ℂ_{F}$ given by the affine model $ℂ_{F} : Y^{l} = F (X)$ , where $F$ is drawn at random uniformly from the set of all monic $l$ -th power-free polynomials $F \in 𝔽_{q} [X]$ of degree $d$ as $d \to \infty$ . The method also enables us to study the fluctuations in the number of $𝔽_{q}$ -points on the same family of curves arising from the set of monic...

An algorithm for extending functions in hypercubes.

Nieto, José Heber (1995)

Divulgaciones Matemáticas

Similarity:

Patterns and periodicity in a family of resultants

Kevin G. Hare, David McKinnon, Christopher D. Sinclair (2009)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Given a monic degree $N$ polynomial $f (x) \in ℤ [x]$ and a non-negative integer $ℓ$ , we may form a new monic degree $N$ polynomial $f_{ℓ} (x) \in ℤ [x]$ by raising each root of $f$ to the $ℓ$ th power. We generalize a lemma of Dobrowolski to show that if $m < n$ and $p$ is prime then $p^{N (m + 1)}$ divides the resultant of $f_{p^{m}}$ and $f_{p^{n}}$ . We then consider the function $(j, k) \mapsto Res (f_{j}, f_{k}) mod p^{m}$ . We show that for fixed $p$ and $m$ that this function is periodic in both $j$ and $k$ , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given. ...