Gropes and the rational lift of the Kontsevich integral

James Conant

Displaying similar documents to “Gropes and the rational lift of the Kontsevich integral”

Various subsemirings of the field $ℚ$ of rational numbers

Vítězslav Kala, Tomáš Kepka, Jon D. Phillips (2009)

Acta Universitatis Carolinae. Mathematica et Physica

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Rational points on curves

Michael Stoll (2011)

Journal de Théorie des Nombres de Bordeaux

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This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve $C$ over $ℚ$ . The focus is on practical aspects of this problem in the case that the genus of $C$ is at least $2$ , and therefore the set of rational points is finite.

A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

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We show that the system of equations $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}$ , where $t_{x} = x (x + 1) / 2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}, t_{x} + t_{y} + t_{z} = t_{s}$ has infinitely many rational two-parameter solutions.

Harmonic functions in four variables with rational and algebraic $p_{4}$ -associates

R. P. Gilbert (1964)

Annales Polonici Mathematici

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Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

Characteristic Exponents of Rational Functions

Anna Zdunik (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider two characteristic exponents of a rational function f:ℂ̂ → ℂ̂ of degree d ≥ 2. The exponent $χ_{a} (f)$ is the average of log∥f’∥ with respect to the measure of maximal entropy. The exponent $χ_{m} (f)$ can be defined as the maximal characteristic exponent over all periodic orbits of f. We prove that $χ_{a} (f) = χ_{m} (f)$ if and only if f(z) is conformally conjugate to $z \mapsto z^{\pm d}$ .

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese, Frédéric Mangolte (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $W \to X$ be a real smooth projective 3-fold fibred by rational curves such that $W (ℝ)$ is orientable. J. Kollár proved that a connected component $N$ of $W (ℝ)$ is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when $X$ is a geometrically rational surface. When $N$ is Seifert fibred over a base orbifold $F$ , our...

Rational solutions of certain Diophantine equations involving norms

Maciej Ulas (2014)

Acta Arithmetica

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We present some results concerning the unirationality of the algebraic variety $_{f}$ given by the equation $N_{K / k} (X ₁ + α X ₂ + α ² X ₃) = f (t)$ , where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and $_{f}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $_{f}$ is k-unirational. A similar result is proved for a broad family of quintic polynomials...

The moduli space of totally marked degree two rational maps

Anupam Bhatnagar (2015)

Acta Arithmetica

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A rational map ϕ: ℙ¹ → ℙ¹ along with an ordered list of fixed and critical points is called a totally marked rational map. The space $R a t ₂^{t} m$ of totally marked degree two rational maps can be parametrized by an affine open subset of (ℙ¹)⁵. We consider the natural action of SL₂ on $R a t ₂^{t} m$ induced from the action of SL₂ on (ℙ¹)⁵ and prove that the quotient space $R a t ₂^{t} m / S L ₂$ exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over ℤ[1/2].

On the Definition of $2$ -Category of $2$ -Knots

V. M. Kharlamov, V. G. Turaev (1993)

Recherche Coopérative sur Programme n°25

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Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$ . We prove Manin’s conjecture on the asymptotic (as $T \to \infty$ ) of the number of $K$ -rational points of $X$ of height less than $T$ , and give an explicit construction of a measure on $X (𝔸)$ , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆 (K)$ on $X (𝔸)$ . Our approach is based on the mixing property of $L^{2} (𝐆 (K) ∖ 𝐆 (𝔸))$ which we obtain with a rate of convergence. ...

On malnormal peripheral subgroups of the fundamental group of a $3$ -manifold

Pierre de la Harpe, Claude Weber (2014)

Confluentes Mathematici

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Let $K$ be a non-trivial knot in the $3$ -sphere, $E_{K}$ its exterior, $G_{K} = π_{1} (E_{K})$ its group, and $P_{K} = π_{1} (\partial E_{K}) \subset G_{K}$ its peripheral subgroup. We show that $P_{K}$ is malnormal in $G_{K}$ , namely that $g P_{K} g^{- 1} \cap P_{K} = {e}$ for any $g \in G_{K}$ with $g \notin P_{K}$ , unless $K$ is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in $E_{K}$ attached to $T_{K}$ which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental...

Springer fiber components in the two columns case for types $A$ and $D$ are normal

Nicolas Perrin, Evgeny Smirnov (2012)

Bulletin de la Société Mathématique de France

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We study the singularities of the irreducible components of the Springer fiber over a nilpotent element $N$ with $N^{2} = 0$ in a Lie algebra of type $A$ or $D$ (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.

A variety of Euler's sum of powers conjecture

Tianxin Cai, Yong Zhang (2021)

Czechoslovak Mathematical Journal

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We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $\{\begin{matrix} n = a_{1} + a_{2} + \dots + a_{s - 1}, \\ a_{1} a_{2} \dots a_{s - 1} (a_{1} + a_{2} + \dots + a_{s - 1}) = b^{s} \end{matrix}$ has positive integer or rational solutions $n$ , $b$ , $a_{i}$ , $i = 1, 2, \dots, s - 1$ , $s \geq 3 .$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s = 3$ , but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s \geq 4$ . As a corollary, for $s \geq 4$ and any positive integer $n$ , the above Diophantine system has a positive rational solution. Meanwhile, we give conditions...

The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner (2011)

Studia Mathematica

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We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_{j}$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n, ν}$ be the space of piecewise linear continuous functions on the torus with knots $t_{j} : 0 \leq j \leq N - 1$ . Finally, let $P_{n, ν}$ be the orthogonal projection operator from L²([0,1)) onto $V_{n, ν}$ . The main result is $l i m_{n \to \infty, ν = 1} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = s u p_{n \in ℕ, 0 \leq ν \leq n} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = 2 + (33 - 18 \sqrt 3) / 13$ . This shows in particular that the Lebesgue constant of the classical...

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Maciej Ulas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let K be a field, a,b ∈ K and ab ≠ 0. Consider the polynomials g₁(x) = xⁿ+ax+b, g₂(x) = xⁿ+ax²+bx, where n is a fixed positive integer. We show that for each k≥ 2 the hypersurface given by the equation $S_{k}^{i} : u ² = \prod_{j = 1}^{k} g_{i} (x_{j})$ , i=1,2, contains a rational curve. Using the above and van de Woestijne’s recent results we show how to construct a rational point different from the point at infinity on the curves $C_{i} : y ² = g_{i} (x)$ , (i=1,2) defined over a finite field, in polynomial time.

The importance of rational extensions

Frans Loonstra (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The rational completion $\bar{M}$ of an $R$ -module $M$ can be characterized as a $\tau_{M}$ -injective hull of $M$ with respect to a (hereditary) torsion functor $\tau_{M}$ depending on $M$ . Properties of a torsion functor depending on an $R$ -module $M$ are studied.