Newton’s method over global height fields

Xander Faber; Adam Towsley

Displaying similar documents to “Newton’s method over global height fields”

On the Newton partially flat minimal resistance body type problems

M. Comte, Jesus Ildefonso Díaz (2005)

Journal of the European Mathematical Society

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We study the flat region of stationary points of the functional $\int_{Ω} F (| \nabla u (x) |) d x$ under the constraint $u \leq M$ , where $Ω$ is a bounded domain in $ℝ^{2}$ . Here $F (s)$ is a function which is concave for $s$ small and convex for $s$ large, and $M > 0$ is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when $Ω$ is a ball. We also analyze some other qualitative properties. Moreover,...

Prime ideal factorization in a number field via Newton polygons

Lhoussain El Fadil (2021)

Czechoslovak Mathematical Journal

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Let $K$ be a number field defined by an irreducible polynomial $F (X) \in ℤ [X]$ and $ℤ_{K}$ its ring of integers. For every prime integer $p$ , we give sufficient and necessary conditions on $F (X)$ that guarantee the existence of exactly $r$ prime ideals of $ℤ_{K}$ lying above $p$ , where $\bar{F} (X)$ factors into powers of $r$ monic irreducible polynomials in $𝔽_{p} [X]$ . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of $ℤ_{K}$ lying above $p$ ....

On sums and products in a field

Guang-Liang Zhou, Zhi-Wei Sun (2022)

Czechoslovak Mathematical Journal

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We study sums and products in a field. Let $F$ be a field with $ch (F) \neq 2$ , where $ch (F)$ is the characteristic of $F$ . For any integer $k \geq 4$ , we show that any $x \in F$ can be written as $a_{1} + \dots + a_{k}$ with $a_{1}, \dots, a_{k} \in F$ and $a_{1} \dots a_{k} = 1$ , and that for any $α \in F ∖ {0}$ we can write every $x \in F$ as $a_{1} \dots a_{k}$ with $a_{1}, \dots, a_{k} \in F$ and $a_{1} + \dots + a_{k} = α$ . We also prove that for any $x \in F$ and $k \in {2, 3, \dots}$ there are $a_{1}, \dots, a_{2 k} \in F$ such that $a_{1} + \dots + a_{2 k} = x = a_{1} \dots a_{2 k}$ .

The centralizer of a classical group and Bruhat-Tits buildings

Daniel Skodlerack (2013)

Annales de l’institut Fourier

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Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G .$ We prove that the Bruhat-Tits building $𝔅^{1} (H)$ of $H$ can be affinely and $G$ -equivariantly embedded in the Bruhat-Tits building $𝔅^{1} (G)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{'}$ be maps from $𝔅^{1} (H)$ to $𝔅^{1} (G)$ which preserve the Moy–Prasad filtrations....

The Golomb space is topologically rigid

Taras O. Banakh, Dario Spirito, Sławomir Turek (2021)

Commentationes Mathematicae Universitatis Carolinae

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The Golomb space $ℕ_{τ}$ is the set $ℕ$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions ${a + b n : n \geq 0}$ with coprime $a, b$ . We prove that the Golomb space $ℕ_{τ}$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.

On the unit group of a semisimple group algebra $𝔽_{q} S L (2, ℤ_{5})$

Rajendra K. Sharma, Gaurav Mittal (2022)

Mathematica Bohemica

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We give the characterization of the unit group of $𝔽_{q} S L (2, ℤ_{5})$ , where $𝔽_{q}$ is a finite field with $q = p^{k}$ elements for prime $p > 5,$ and $S L (2, ℤ_{5})$ denotes the special linear group of $2 \times 2$ matrices having determinant $1$ over the cyclic group $ℤ_{5}$ .

On the derived length of units in group algebra

Dishari Chaudhuri, Anupam Saikia (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group $G$ , $K$ a field of characteristic $p \geq 17$ and let $U$ be the group of units in $K G$ . We show that if the derived length of $U$ does not exceed $4$ , then $G$ must be abelian.

On monogenity of certain pure number fields of degrees $2^{r} \cdot 3^{k} \cdot 7^{s}$

Hamid Ben Yakkou, Jalal Didi (2024)

Mathematica Bohemica

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Let $K = ℚ (α)$ be a pure number field generated by a complex root $α$ of a monic irreducible polynomial $F (x) = x^{2^{r} \cdot 3^{k} \cdot 7^{s}} - m \in ℤ [x]$ , where $r$ , $k$ , $s$ are three positive natural integers. The purpose of this paper is to study the monogenity of $K$ . Our results are illustrated by some examples.

An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $C$ be a hyperelliptic curve of genus $g \geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$ . We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$ , $S$ and $K$ . In particular, we obtain that for any given number field $K$ , finite set of places $S$ of $K$ and integer $g \geq 1$ one can in principle determine the set of $K$ -isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

Purity of level $m$ stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let $k$ be a field of characteristic $p > 0$ . Let $D_{m}$ be a ${BT}_{m}$ over $k$ (i.e., an $m$ -truncated Barsotti–Tate group over $k$ ). Let $S$ be a $k$ -scheme and let $X$ be a ${BT}_{m}$ over $S$ . Let $S_{D_{m}} (X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_{m}$ . If $p \geq 5$ , we show that $S_{D_{m}} (X)$ is pure in $S$ , i.e. the immersion $S_{D_{m}} (X) ↪ S$ is affine. For $p \in {2, 3}$ , we prove purity if $D_{m}$ satisfies a certain technical property depending only on its $p$ -torsion $D_{m} [p]$ . For $p \geq 5$ , we apply the developed techniques to show that...

A density version of the Carlson–Simpson theorem

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros (2014)

Journal of the European Mathematical Society

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We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k \geq 2$ and every set $A$ of words over $k$ satisfying $\lim \sup_{n \to \infty} | A \cap {[k]}^{n} | / k^{n} > 0$ there exist a word $c$ over $k$ and a sequence $(w_{n})$ of left variable words over $k$ such that the set $c \cup {c^{} w_{0} {(a_{0})}^{} . . .^{} w_{n} (a_{n}) : n \in ℕ and a_{0}, . . ., a_{n} \in [k]}$ is contained in $A$ . While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.