Thomas Scanlon

University of California, Berkeley | UCB · Department of Mathematics

Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a prerequisite for meaningful parameter identification in practice. In the presence of nonidentifiability, it is important to find all functions of the parameters that...

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the siz...

Structural identifiability is a property of an ODE model with parameters that allows for the parameters to be determined from continuous noise-free data. This is natural prerequisite for practical identifiability. Conducting multiple independent experiments could make more parameters or functions of parameters identifiable, which is a desirable pro...

We establish effective elimination theorems for ordinary differential-difference equations. Specifically, we find a computable function B ( r , s ) B(r,s) of the natural number parameters r r and s s so that for any system of algebraic ordinary differential-difference equations in the variables x = x 1 , … , x q \mathbfit {x} = x_1, \ldots , x_q an...

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the un...

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the siz...

Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a prerequisite for meaningful parameter identification in practice. In the presence of nonidentifiability, it is important to find all functions of the parameters that...

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (e.g., standard difference schemes) and difference equations in functions on words. On the universal...

We establish effective elimination theorems for differential-difference equations. Specifically, we find a computable function $B(r,s)$ of the natural number parameters $r$ and $s$ so that for any system of algebraic differential-difference equations in the variables $\mathbf{x} = x_1, \ldots, x_q$ and $\mathbf{y} = y_1, \ldots, y_r$ each of which...

The Mordell-Lang conjecture (proven by Faltings, Vojta and McQuillan) states that the intersection of a subvariety V of a semiabelian variety G defined over an algebraically closed field K of characteristic 0 with a finite rank subgroup U is a finite union of cosets of subgroups of U. We explore a variant of this conjecture when G is a product of a...

Let K be an algebraically closed field of prime characteristic p, let X be a semiabelian variety defined over a finite subfield of K, let f be a regular self-map on X defined over K, let V be a subvariety of X defined over K, and let x be a K-point of X. The Dynamical Mordell-Lang Conjecture in characteristic p predicts that the set S consisting of...

We prove effective Nullstellensatz and elimination theorems for difference equations in sequence rings. More precisely, we compute an explicit function of geometric quantities associated to a system of difference equations (and these geometric quantities may themselves be bounded by a function of the number of variables, the order of the equations,...

We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring $A$ is bi-interpretable with $(\mathbb N,{+},{\times})$ if and only if the space of non-maximal prime ideals of $A$ is nonempty and connected in the Zariski topology and the nilradical of $A$...

H\"older-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of $L^p$ norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suita...

Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of Gao, Li and Wei. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of character...

Let $A$ be an abelian variety defined over $\bar{\mathbb{Q}}$, and let $\varphi$ be a dominant endomorphism of $A$ as an algebraic variety. We prove that either there exists a non-constant rational fibration preserved by $\varphi$, or there exists a point $x\in A(\bar{\mathbb{Q}})$ whose $\varphi$-orbit is Zariski dense in $A$. This provides a posi...

Let $Y$ be a complex algebraic variety, $G \curvearrowright Y$ an action of an algebraic group on $Y$, $U \subseteq Y({\mathbb C})$ a complex submanifold, $\Gamma < G({\mathbb C})$ a discrete, Zariski dense subgroup of $G({\mathbb C})$ which preserves $U$, and $\pi:U \to X({\mathbb C})$ an analytic covering map of the complex algebraic variety $X$...

We show that the order three algebraic differential equation over ${\mathbb Q}$ satisfied by the analytic $j$-function defines a non-$\aleph_0$-categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of suc...

The movement of data (communication) between levels of a memory hierarchy, or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication are of interest. Motivated by this, attainable lower bounds for the amount of communication required by algorithms were established by several...

Communication, i.e., moving data, between levels of a memory hierarchy or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication can run much faster (and use less energy) than algorithms that do not. Motivated by this, attainable communication lower bounds were established i...

Generalising and unifying the known theorems for difference and differential fields, it is shown that for every finite free ${\mathbb S}$-algebra ${\mathcal D}$ over a field $A$ of characteristic zero the theory of ${\mathcal D}$-fields has a model companion ${\mathcal D}$-CF$_0$ which is simple and satisfies the Zilber dichotomy for finite-dimensi...

We show that for each finite sequence of algebraic integers $\alpha_1,...,\alpha_n$ and polynomials $P_1(x_1,...,x_n;y_1,...,y_n),..., P_r(x_1,...,x_n;y_1,...,y_n)$ with algebraic integer coefficients, there are a natural number $N$, $n$ commuting endomorphisms $\Phi_i:\Gm^N \to \Gm^N$ of the $N^\text{th}$ Cartesian power of the multiplicative grou...

We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier.

Let $K$ be a number field, let $\phi \in K(t)$ be a rational map of degree at least 2, and let $\alpha, \beta \in K$. We show that if $\alpha$ is not in the forward orbit of $\beta$, then there is a positive proportion of primes ${\mathfrak p}$ of $K$ such that $\alpha \mod {\mathfrak p}$ is not in the forward orbit of $\beta \mod {\mathfrak p}$. M...

We show that dependent elds have no Artin-Schreier extension, and that simple elds have only a nite number of them.

Let $k$ be a field of positive characteristic and $K = k(V)$ a function field of a variety $V$ over $k$ and let ${\mathbf A}_K$ be a ring of ad\'{e}les of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $\Phi:{\mathbb F}[t] \to \operatorname{End}_K({\mathbb G}_a)$ over $K$ and a...

Using the theory of o-minimality we show that the $p$-adic method of Skolem-Mahler-Lech-Chabauty may be adapted to prove instances of the dynamical Mordell-Lang conjecture for some real analytic dynamical systems. For example, we show that if $f_1,...,f_n$ is a finite sequence of real analytic functions $f_i:(-1,1) \to (-1,1)$ for which $f_i(0) = 0...

We show that NIP fields have no Artin-Schreier extension, and that simple fields have only a finite number of them.

Building on the abstract notion of prolongation developed by Moosa and Scanlon ('Jet and prolongation spaces', J. Inst. Math. Jussieu 9 (2010), 391–430), the theory of iterative Hasse–Schmidt rings and schemes is introduced, simultaneously generalizing difference and (Hasse–Schmidt) differential rings and schemes. This work provides a unified forma...

We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$ given by coordinatewise univariate polynomials by refining a theorem of Ritt. More precisely, we find a nearly canonical way to write a polynomial as a composition of "clusters". Our main result is an explici...

Let $(K,|\cdot|)$ be a complete discretely valued field and $f:{\mathbb B}_1(K,1) \to {\mathbb B}_1(K,1)$ a nonconstant analytic map from the unit back to itself. We assume that 0 is an attracting fixed point of $f$. Let $a \in K$ with $\lim_{n \to \infty} f^n(a) = 0$ and consider the orbit ${\mathcal O}_f(a) := \{f^n(a) : n \in {\mathbb N} \}$. We...

The notion of prolongation of an algebraic variety is developed in an abstract setting that generalises the difference and (Hasse) differential contexts. An interpolating map that compares the prolongation spaces with algebraic jet spaces is introduced and studied.

We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point. Using differential arcs, we show that if (K, +, ×, δ 1 , . . . , δn) is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then e...

Using the work of several other mathematicians, principally the results of Poonen refining the work of Pop that algebraic independence is definable within the class of finitely generated fields and of Rumely that the ring of rational integers is uniformly interpreted in global fields, and a theorem on the definability of valuations on function fiel...

In answer to a question of L. van den Dries, we show that no differentially closed field possesses a differential valuation.

Abstract: We give axiomatizations and prove quantifier elimination theorems for first-order theories of unramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use t...

We prove a p-adic version of the André-Oort conjecture for subvarieties of the universal abelian varieties. Let g and n be integers with n≥3 and p a prime number not dividing n. Let R be a finite extension of \(W[\mathbb{F}_{p}^{alg}]\), the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space \(\mathcal{A}=\ma...

We characterise strongly minimal groups interpretable in elementary extensions of compact complex analytic spaces.

Extending the work of [A. Pillay, T. Scanlon, Meromorphic groups, Trans. AMS 355 (10) (2003) 3843–3859] on groups definable in compact complex manifolds and of [M. Aschenbrenner, R. Moosa, T. Scanlon, Groups in the theory of compact complex spaces, 2005] on strongly minimal groups definable in nonstandard compact complex manifolds, we classify all...

Generalizing and synthesizing earlier work on the model theory of valued difference fields and on the model theory of valued fields with analytic structure, we prove Ax�Kochen� Er�ov style relative completeness and relative quantifier elimination theorems for a theory of valuation rings with analytic and difference structure. Specializing our resul...

Let X be an algebraic variety over the algebraically closed field K and Ξ ⊆ X(K) a set of K-rational points on X. We say that a subvariety Y ⊆ Xn of some Cartesian power of X is Ξ-special if Ξn ∩ Y(K) is Zariski dense in Y. We show under certain hypotheses on Ξ, for instance, that the class of Ξ-special varieties is closed under intersections, that...

We present the details of a model theoretic proof of an analogue of the Manin-Mumford conjecture for semiabelian varieties in positive characteristic.

Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination and stability result for finitely generated modules over certain finite simple extensions of the integers given together with predicates for cycles of the distinguished gener...

This paper stems from my lecture notes for a talk given at Rutgers University in Newark on 3 November 2000 as part of the Workshop on Dierential Algebra and Related Topics. I thank Li Guo for inviting me to speak and for organizing such a successful meeting of the disparate strands of the dierential algebra community. I thank also the referee for c...

We show that analogues of popular public key cryptosystems based on Drinfeld modules are insecure by providing polynomial time algorithms to solve the Drinfeld module versions of the inversion and discrete logarithm problems.

We note that Pillay's result on the stability of an algebraically closed field with a predicate for a group of Lang type implies that number uniformity follows formally from the finiteness results analogous to Faltings' Theorem.

We point out that a certain complex compact manifold constructed by Lieberman has the dimensional order property, and has U-rank di#erent from Morley rank. We also give a su#cient condition for a Kahler manifold to be totally degenerate (that is, to be an indiscernible set, in its canonical language) and point out that there are K3 surfaces which s...

§ 1. Introduction . With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument maske...

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generaliza...

Introduction and Background The note contains the details of an assertion made in [1] to the eect that elds admitting a nontrivial strong ordered Euler characteristic are quasinite. In this section we recall the relevant denitions and in the next section we complete the proof. Recall that a eld K is quasinite if K is perfect and its absolute Galois...

We show that a connected group interpretable in a compact complex manifold (a meromorphic group) is definably an extension of a complex torus by a linear algebraic group, generalizing results in [4]. A special case of this result, as well as one of the ingredients in the proof, is that a strongly minimal modular meromorphic group is a complex torus...

. We describe intersections of finitely generated subgroups of semi-abelian varieties with subvarieties in characteristic p. 1. Introduction A version of the Mordell-Lang Conjecture in characteristic zero asserts that if G is a semi-abelian variety, # # G is a finitely generated group, and X # G is a subvariety, then the Zariski closure of X # # is...

. Tate and Voloch have conjectured that the p-adic distance from torsion points of semi-abelian varieties over C p to subvarieties may be uniformly bounded. We prove this conjecture for prime-to-p torsion points on semi-abelian varieties over Q alg p using methods of algebraic model theory. Let C p denote the completion of the algebraic closure of...

It is proved that any supersimple eld has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and elds whose theory is simple. 1 Introduction Simple theories were introduced by Shelah in [12]. In [3], Kim, continuing Shelah's work, showed...

. Tate and Voloch have conjectured that the p-adic distance from torsion points of semi-abelian varieties over C p to subvarieties may be uniformly bounded. We prove this conjecture for torsion points on semi-abelian varieties over Qp alg using methods of algebraic model theory and a result of Sen on Galois representation of Hodge-Tate type. As a g...

Buium proved what he called the abc theorem for abelian varieties over function fields in characteristic zero [3]. Using methods of algebraic model theory we prove an analog of his theorem for commutative algebraic groups in characteristic p.

In this thesis we introduce a general notion of a D-ring generalizing that of a differential or difference ring. In Chapter 3, this notion is specialized to consider valued fields D-fields: valued fields K having an operator D : K ! K and a fixed element e 2 K satisfying D(x + y) = Dx + Dy, D(1) = 0, D(xy) = xDy + yDx + eDxDy, v(e) 0, and v(Dx) v(x...

. We note that if K is an innite stable eld of characteristic p > 0, then the Artin-Schreier map } : K ! K given by x 7! x p x is surjective. Consequently, K has no nite Galois extensions of degree divisible by p. An old theorem of Macintyre asserts that an !-stable innite eld is algebraically closed [4]. Cherlin and Shelah generalized this theorem...

. We prove an analogue of the Manin-Mumford conjecture for Drinfeld modules of generic characteristic. 1. Introduction L. Denis proposed that the qualitative diophantine results known for certain subgroups of semi-abelian varieties should hold for Drinfeld modules. In particular, the Manin-Mumford conjecture which asserts that an irreducible subvar...

. Let (K;v) be a complete discretely valued field of characteristic zero with an algebraically closed residue field of positive characteristic. Let oe : K ! K be a continuous automorphism of K inducing a Frobenius automorphism on the residue field. We prove quantifierelimination for (K;v; oe) in a language with angular component maps and in a langu...

. The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the AxKochen -Ershov principle is proven for a theory of valued D-fields of residual characteristic zero. The model theory of differential and difference fields has been extensively studied (see for example [7,...

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference cl...

It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.

We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families. We will be interested here only in sets of finite Morley rank. Section 1 consists of some general lemmas...

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially or- dered Grothendieck ring. We give a generali...

Let p be a prime number and f 2 Zp(x) a polynomial over the p-adic integers lifting the Frobenius in the sense that f(x) xp (mod p) and deg(f) = p. Let f := { 2 Q alg p : f n( ) = for some n 2 Z+} be the set of f-periodic points. We show that an irreducible algebraic varieties X Am which meet ◊m f in a Zariski dense set only in the case that X is d...