Joseph H. Silverman
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View article: Maximizing Satisfied Vertex Requests in List Coloring
Maximizing Satisfied Vertex Requests in List Coloring Open
Suppose $G$ is a graph and $L$ is a list assignment for $G$. A request of $L$ is a function $r$ with nonempty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$. The triple $(G,L,r)$ is $ε$-satisfiable if there exists a …
View article: Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions
Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions Open
In this note we give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system f:X-->X, including the dynamical degree d(f), which gives a coarse measure of the geometric complexity of the i…
View article: A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over Function Fields
A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over Function Fields Open
Let $\mathbb{F}$ be the function field of a curve over an algebraically closed field with $\operatorname{char}(\mathbb{F})\ne2,3$, and let $E/\mathbb{F}$ be an elliptic curve. Then for all finite extensions $\mathbb{K}/\mathbb{F}$ and all …
View article: Survey lecture on arithmetic dynamics
Survey lecture on arithmetic dynamics Open
Arithmetic dynamics is a relatively new field in which classical problems from number theory and algebraic geometry are reformulated in the setting of dynamical systems. Thus, for example, rational points on algebraic varieties become rati…
View article: The size of semigroup orbits modulo primes
The size of semigroup orbits modulo primes Open
Let V be a projective variety defined over a number field K , let S be a polarized set of endomorphisms of V all defined over K , and let P ∈ V (K ).For each prime p of K , let m p (S, P) denote the number of points in the orbit of P mod p…
View article: Propagation of Zariski Dense Orbits
Propagation of Zariski Dense Orbits Open
Let $X/K$ be a smooth projective variety defined over a number field, and let $f:X\to{X}$ be a morphism defined over $K$. We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point $P_0…
View article: The size of semigroup orbits modulo primes
The size of semigroup orbits modulo primes Open
Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $P\in V(K)$. For each prime $\mathfrak{p}$ of $K$, let $m_{\mathfrak{p}}(S,P)$ denote the num…
View article: A Heuristic Subexponential Algorithm to Find Paths in Markoff Graphs Over Finite Fields
A Heuristic Subexponential Algorithm to Find Paths in Markoff Graphs Over Finite Fields Open
Charles, Goren, and Lauter [J. Cryptology 22(1), 2009] explained how one can construct hash functions using expander graphs in which it is hard to find paths between specified vertices. The set of solutions to the classical Markoff equatio…
View article: Orbits on K3 Surfaces of Markoff Type
Orbits on K3 Surfaces of Markoff Type Open
Let $\mathcal{W}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be a surface given by the vanishing of a $(2,2,2)$-form. These surfaces admit three involutions coming from the three projections $\mathcal{W}\to\mathbb{P}^1\times\ma…
View article: A Lehmer-type height lower bound for abelian surfaces over function fields
A Lehmer-type height lower bound for abelian surfaces over function fields Open
Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(\bar{K})$. More precisely,…
View article: A uniform quantitative Manin-Mumford theorem for curves over function fields
A uniform quantitative Manin-Mumford theorem for curves over function fields Open
We prove that any smooth projective geometrically connected non-isotrivial curve of genus $g\ge 2$ over a one-dimensional function field of any characteristic has at most $16g^2+32g+124$ torsion points for any Abel--Jacobi embedding of the…
View article: The Distribution Relation and Inverse Function Theorem in Arithmetic Geometry
The Distribution Relation and Inverse Function Theorem in Arithmetic Geometry Open
We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis on versions that can be applied uniformly across families of varieties and maps. In particular, we prove two…
View article: A signature scheme from the finite field isomorphism problem
A signature scheme from the finite field isomorphism problem Open
In a recent paper the authors and their collaborators proposed a new hard problem, called the finite field isomorphism problem, and they used it to construct a fully homomorphic encryption scheme. In this paper, we investigate how one migh…
View article: GIT stability of Hénon maps
GIT stability of Hénon maps Open
In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space $\operatorname{Rat}_d^N$ of degree $d$ rational maps $\mathbb{P}^N\to\mathbb{P}^N$ modulo the conjugation action of $\operatorname{SL}_{N+1}$. We …
View article: The Judeo-Spanish Ballad Tradition
The Judeo-Spanish Ballad Tradition Open
As the repertoire of an isolated, archaizing minority, the ballad tradition of the Spanish-speaking Sephardic Jews constitutes one of the most distinctive and interesting branches of the Hispanic romancero. The first full text of a Sephard…
View article: A signature scheme from the finite field isomorphism problem
A signature scheme from the finite field isomorphism problem Open
In a recent paper the authors and their collaborators proposed a new hard problem, called the finite field isomorphism problem , and they used it to construct a fully homomorphic encryption scheme. In this paper, we investigate how one mig…
View article: Post-Critically Finite Maps on $\mathbb{P}^n$ for $n\ge2$ are Sparse
Post-Critically Finite Maps on $\mathbb{P}^n$ for $n\ge2$ are Sparse Open
Let $f:{\mathbb P}^n\to{\mathbb P}^n$ be a morphism of degree $d\ge2$. The map $f$ is said to be post-critically finite (PCF) if there exist integers $k\ge1$ and $\ell\ge0$ such that the critical locus $\operatorname{Crit}_f$ satisfies $f^…
View article: Addendum to “Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties"
Addendum to “Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties" Open
The authors note that certain height formulas in their paper had been proven earlier by Bertrand.
View article: Current trends and open problems in arithmetic dynamics
Current trends and open problems in arithmetic dynamics Open
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry and partly from…
View article: To Write or Not to Write...A Book, and When?
To Write or Not to Write...A Book, and When? Open
feel that you have something to say.And if that's the case, don't spend time worrying about general criticism from people who tell you about the book that you "should" be writing.As Dot admonishes Seurat in Sondheim's Sunday in the park wi…
View article: Multiplicative dependence among iterated values of rational functions modulo finitely generated groups
Multiplicative dependence among iterated values of rational functions modulo finitely generated groups Open
We study multiplicative dependence between elements in orbits ofalgebraic dynamical systems over number fields modulo a finitely generated multiplicative subgroup of the field. We obtain a series of results, many of which may be viewed as …
View article: Integrality properties of Böttcher coordinates for one-dimensional superattracting germs
Integrality properties of Böttcher coordinates for one-dimensional superattracting germs Open
Let $R$ be a ring of characteristic $0$ with field of fractions $K$ and let $m\geq 2$ . The Böttcher coordinate of a power series $\unicode[STIX]{x1D711}(x)\in x^{m}+x^{m+1}R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$ is the unique power …
View article: A Uniform Field-of-Definition/Field-of-Moduli Bound for Dynamical Systems on $\mathbf{P}^N$
A Uniform Field-of-Definition/Field-of-Moduli Bound for Dynamical Systems on $\mathbf{P}^N$ Open
Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be an endomorphism of degree $d\ge2$ defined over $\overline{\mathbb{Q}}$ or $\overline{\mathbb{Q}}_p$, and let $K$ be the field of moduli of $f$. We prove that there is a field of definition $L$ for $f$…
View article: Degeneration of dynamical degrees in families of maps
Degeneration of dynamical degrees in families of maps Open
The dynamical degree of a dominant rational map\n$f:\\mathbb{P}^N\\rightarrow\\mathbb{P}^N$ is the quantity\n$\\delta(f):=\\lim(\\text{deg} f^n)^{1/n}$. We study the variation of dynamical\ndegrees in 1-parameter families of maps $f_T$. We…
View article: Integrality properties of B\\"ottcher coordinates for one-dimensional\n superattracting germs
Integrality properties of B\\"ottcher coordinates for one-dimensional\n superattracting germs Open
Let $R$ be a ring of characteristic $0$ with field of fractions $K$, and let\n$m\\ge2$. The B\\"ottcher coordinate of a power series $\\varphi(x)\\in x^m +\nx^{m+1}R[\\![x]\\!]$ is the unique power series $f_\\varphi(x)\\in x+x^2K[\\![x]\\…
View article: Good reduction of dynamical systems with preperiodic level structure and a Shafarevich-type conjecture
Good reduction of dynamical systems with preperiodic level structure and a Shafarevich-type conjecture Open
Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:\mathbb{P}^N_K\to\mathbb{P}^N_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morph…
View article: Rational points on, and the arithmetic of, elliptic curves: A tale of two books (and an article)
Rational points on, and the arithmetic of, elliptic curves: A tale of two books (and an article) Open
It was the best of talks, it was the worst of talks,Our tale begins in 1961, when Professor John Tate was invited by John Solomon to deliver a series of lectures 1 at Haverford College on the subject of "Rational Points on Cubic Curves" [8…
View article: Finite ramification for preimage fields of post-critically finite morphisms
Finite ramification for preimage fields of post-critically finite morphisms Open
Given a finite endomorphism $φ$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(φ^{-\infty}(α)) : = \bigcup_{n \geq 1} K(φ^{-n}(α))$ generated by the preimages of $α$ under all itera…