scholarly journals Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable

2014 ◽  
Vol 36 (4) ◽  
pp. 1156-1166 ◽  
Author(s):  
IGORS GORBOVICKIS
Keyword(s):  
Periodic Orbits ◽  
Moduli Space ◽  
Algebraic Independence ◽  
Complex Polynomials ◽  
Generic Point ◽  
Algebraic Functions ◽  
Polynomial Maps ◽  
Affine Subspaces

We consider the space of complex polynomials of degree $n\geq 3$ with $n-1$ distinct marked periodic orbits of given periods. We prove that this space is irreducible and the multipliers of the marked periodic orbits, considered as algebraic functions on that space, are algebraically independent over $\mathbb{C}$. Equivalently, this means that at its generic point the moduli space of degree-$n$ polynomial maps can be locally parameterized by the multipliers of $n-1$ arbitrary distinct periodic orbits. We also prove a similar result for a certain class of affine subspaces of the space of complex polynomials of degree $n$.

scholarly journals Arithmetic of Unicritical Polynomial Maps

2014 ◽  
Author(s):  
John Milnor
Keyword(s):  
Normal Form ◽  
Critical Point ◽  
Periodic Orbits ◽  
Integral Closure ◽  
Complex Polynomial ◽  
Complex Polynomials ◽  
Arithmetic Properties ◽  
Polynomial Maps ◽  
Finite Case ◽  
External Rays

This chapter studies complex polynomials with only one critical point, relating arithmetic properties of the coefficients to those of periodic orbits and their multipliers and external rays. It first defines the complex polynomial maps of degree n ≥ 2, and draws an alternate normal form for studying periodic orbits. The chapter also discusses the notation for the integral closure. Next, the chapter discusses several statements about periodic orbits. It then proceeds to lay out the proofs of these statements, in the process detailing some basic properties of the integral closure. Finally, the chapter closes with a discussion of the critically finite case.

scholarly journals Lifting 1/4-BPS states in AdS3× S3 × T4

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Nathan Benjamin ◽  
Christoph A. Keller ◽  
Ida G. Zadeh
Keyword(s):  
Perturbation Theory ◽  
Moduli Space ◽  
Second Order ◽  
Generic Point ◽  
Orbifold Point ◽  
Bps States

Abstract We establish a framework for doing second order conformal perturbation theory for the symmetric orbifold SymN(T4) to all orders in N. This allows us to compute how 1/4-BPS states of the D1-D5 system on AdS3 × S3 × T4 are lifted as we move away from the orbifold point. As an application we confirm a previous observation that in the large N limit not all 1/4-BPS states that can be lifted do get lifted. This provides evidence that the supersymmetric index actually undercounts the number of 1/4-BPS states at a generic point in the moduli space.

scholarly journals Microstate geometries at a generic point in moduli space

2019 ◽  
Vol 51 (9) ◽  
Author(s):  
Guillaume Bossard ◽  
Severin Lüst
Keyword(s):  
Moduli Space ◽  
Generic Point

scholarly journals The ring of algebraic correspondences on a generic curve of genus g

1976 ◽  
Vol 60 ◽  
pp. 173-180 ◽  
Author(s):  
Yoshiaki Ikeda
Keyword(s):  
Prime Number ◽  
Moduli Space ◽  
Algebraic Closure ◽  
Negative Integer ◽  
Generic Point ◽  
Prime Field ◽  
Irreducible Variety ◽  
Generic Curve

0. Let p be a prime number or zero and let g be a non-negative integer. Then there is a coarse moduli space Mg for complete non-singular irreducible curves of genus g defined over fields of characteristic p, which is an irreducible variety over the algebraic closure F̅p of the prime field Fp. (Especially, F0 is also denoted by Q as usual.) ([8], [2]). The curve corresponding to a generic point of Mg over F̅p is called a generic curve of genus g.

scholarly journals The tadpole problem

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Iosif Bena ◽  
Johan Blåbäck ◽  
Mariana Graña ◽  
Severin Lüst
Keyword(s):  
Moduli Space ◽  
Complex Structure ◽  
De Sitter ◽  
Generic Point ◽  
Cancellation Condition ◽  
Moduli Stabilization ◽  
Induced Charge ◽  
De Sitter Vacua ◽  
Breaking Scale ◽  
Rule Out

Abstract We examine the mechanism of moduli stabilization by fluxes in the limit of a large number of moduli. We conjecture that one cannot stabilize all complex-structure moduli in F-theory at a generic point in moduli space (away from singularities) by fluxes that satisfy the bound imposed by the tadpole cancellation condition. More precisely, while the tadpole bound in the limit of a large number of complex-structure moduli goes like 1/4 of the number of moduli, we conjecture that the amount of charge induced by fluxes stabilizing all moduli grows faster than this, and is therefore larger than the allowed amount. Our conjecture is supported by two examples: K3 × K3 compactifications, where by using evolutionary algorithms we find that moduli stabilization needs fluxes whose induced charge is 44% of the number of moduli, and Type IIB compactifications on $$ \mathbbm{CP} $$ CP 3, where the induced charge of the fluxes needed to stabilize the D7-brane moduli is also 44% of the number of these moduli. Proving our conjecture would rule out de Sitter vacua obtained via antibrane uplift in long warped throats with a hierarchically small supersymmetry breaking scale, which require a large tadpole.

Typical properties of periodic Teichmüller geodesics: Lyapunov exponents

2021 ◽  
pp. 1-29
Author(s):  
URSULA HAMENSTÄDT
Keyword(s):  
Growth Rate ◽  
Vector Bundle ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Moduli Space ◽  
Abelian Differentials ◽  
The Matrix ◽  
Teichmüller Flow

Abstract Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

scholarly journals THE p-TORSION OF CURVES WITH LARGE p-RANK

2009 ◽  
Vol 05 (06) ◽  
pp. 1103-1116 ◽  
Author(s):  
RACHEL PRIES
Keyword(s):  
Moduli Space ◽  
Open Problem ◽  
Hyperelliptic Curve ◽  
Closed Field ◽  
Generic Point ◽  
Characteristic P ◽  
Smooth Curves ◽  
Algebraically Closed Field ◽  
Group Schemes

Consider the moduli space of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for f < g - 1. We prove that the generic point of every component of this moduli space has a-number 1 when f = g - 2 and f = g - 3. Likewise, we show that a generic hyperelliptic curve with p-rank g - 2 has a-number 1 when p ≥ 3. We also show that the locus of curves with p-rank g - 2 and a-number 2 is non-empty with codimension 3 in [Formula: see text] when p ≥ 5. We include some other results when f = g - 3. The proofs are by induction on g while fixing g - f. They use computations about certain components of the boundary of [Formula: see text].

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