Density of rational points on cyclic covers of $\mathbb{P}^n$

Ritabrata Munshi

Displaying similar documents to “Density of rational points on cyclic covers of $ℙ^{n}$ ”

A generalization of Scholz’s reciprocity law

Mark Budden, Jeremiah Eisenmenger, Jonathan Kish (2007)

Journal de Théorie des Nombres de Bordeaux

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We provide a generalization of Scholz’s reciprocity law using the subfields $K_{2^{t - 1}}$ and $K_{2^{t}}$ of $ℚ (ζ_{p})$ , of degrees $2^{t - 1}$ and $2^{t}$ over $ℚ$ , respectively. The proof requires a particular choice of primitive element for $K_{2^{t}}$ over $K_{2^{t - 1}}$ and is based upon the splitting of the cyclotomic polynomial $Φ_{p} (x)$ over the subfields.

Counting monic irreducible polynomials $P$ in $𝔽_{q} [X]$ for which order of $X (mod P)$ is odd

Christian Ballot (2007)

Journal de Théorie des Nombres de Bordeaux

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Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2 (mod p)$ is odd; it is $7 / 24$ . Here we mimic successfully Hasse’s method to compute the density $δ_{q}$ of monic irreducibles $P$ in $𝔽_{q} [X]$ for which the order of $X (mod P)$ is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the $δ_{p}$ ’s as $p$ varies through all rational primes.

On the parity of generalized partition functions, III

Fethi Ben Saïd, Jean-Louis Nicolas, Ahlem Zekraoui (2010)

Journal de Théorie des Nombres de Bordeaux

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Improving on some results of J.-L. Nicolas [], the elements of the set $𝒜 = 𝒜 (1 + z + z^{3} + z^{4} + z^{5})$ , for which the partition function $p (𝒜, n)$ (i.e. the number of partitions of $n$ with parts in $𝒜$ ) is even for all $n \geq 6$ are determined. An asymptotic estimate to the counting function of this set is also given.

On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression

Maurizio Laporta (2012)

Journal de Théorie des Nombres de Bordeaux

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We prove a Bombieri-Vinogradov type theorem for the number of representations of an integer $N$ in the form $N = p_{1}^{g} + p_{2}^{g} + ... + p_{s}^{g}$ with $p_{1}, p_{2}, ..., p_{s}$ prime numbers such that $p_{1} \equiv l (\mod k)$ , under suitable hypothesis on $s = s (g)$ for every integer $g \geq 2$ .

Practical Aurifeuillian factorization

Bill Allombert, Karim Belabas (2008)

Journal de Théorie des Nombres de Bordeaux

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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 $L ≔ log (|a| + 1)$ .

Displaying similar documents to “Density of rational points on cyclic covers of ℙ n ”

A generalization of Scholz’s reciprocity law

Counting monic irreducible polynomials P in 𝔽 q [ X ] for which order of X ( mod P ) is odd

On the parity of generalized partition functions, III

On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression

Practical Aurifeuillian factorization

Displaying similar documents to “Density of rational points on cyclic covers of $ℙ^{n}$ ”

Counting monic irreducible polynomials $P$ in $𝔽_{q} [X]$ for which order of $X (mod P)$ is odd