James Freitag
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View article: FINITE-DIMENSIONAL DIFFERENTIAL-ALGEBRAIC PERMUTATION GROUPS
FINITE-DIMENSIONAL DIFFERENTIAL-ALGEBRAIC PERMUTATION GROUPS Open
Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably prim…
View article: Simple Homogeneous Structures and Indiscernible Sequence Invariants
Simple Homogeneous Structures and Indiscernible Sequence Invariants Open
We introduce some properties describing dependence in indiscernible sequences: $F_{ind}$ and its dual $F_{Mb}$, the definable Morley property, and $n$-resolvability. Applying these properties, we establish the following results: We show th…
View article: Applications of Littlestone Dimension to Query Learning and to Compression
Applications of Littlestone Dimension to Query Learning and to Compression Open
In this paper we give several applications of Littlestone dimension. The first is to the model of [Angluin and Dohrn, 2017], where we extend their results for learning by equivalence queries with random counterexamples. Second, we extend t…
View article: Bounding nonminimality and a conjecture of Borovik–Cherlin
Bounding nonminimality and a conjecture of Borovik–Cherlin Open
Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of non…
View article: Order one differential equations on nonisotrivial algebraic curves
Order one differential equations on nonisotrivial algebraic curves Open
In this paper we provide new examples of geometrically trivial strongly minimal differential algebraic varieties living on nonisotrivial curves over differentially closed fields of characteristic zero. Our technique involves developing a t…
View article: Finite-dimensional differential-algebraic permutation groups
Finite-dimensional differential-algebraic permutation groups Open
Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably prim…
View article: Generic differential equations are strongly minimal
Generic differential equations are strongly minimal Open
In this paper we develop a new technique for showing that a nonlinear algebraic differential equation is strongly minimal based on the recently developed notion of the degree of non-minimality of Freitag and Moosa. Our techniques are suffi…
View article: On the Geometry of Stable Steiner Tree Instances
On the Geometry of Stable Steiner Tree Instances Open
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View article: The degree of nonminimality is at most two
The degree of nonminimality is at most two Open
It is shown that if $p$ is a complete type of Lascar rank at least 2 over $A$, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations, $a_1$ and $a_2$, such that $p$ has a nonalgebrai…
View article: Strong minimality of triangle functions
Strong minimality of triangle functions Open
In this manuscript, we give a new proof of strong minimality of certain automorphic functions, originally results of Freitag and Scanlon (2017), Casale, Freitag, and Nagloo (2020), Blázquez-Sanz, Casale, Freitag, and Nagloo (2020). Our pro…
View article: On the equations of Poizat and Liénard
On the equations of Poizat and Liénard Open
We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Li\'enard equations. We give a necessary and sufficient condit…
View article: When any three solutions are independent
When any three solutions are independent Open
Given an algebraic differential equation of order greater than one, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already s…
View article: Not Pfaffian
Not Pfaffian Open
This short note describes the connection between strong minimality of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being Pfaffian. This connection combined with a theo…
View article: Not Pfaffian.
Not Pfaffian. Open
This short note describes the connection between strong minimality of the
differential equation satisfied by an complex analytic function and the real
and imaginary parts of the function being Pfaffian. This connection combined
with a theo…
View article: Bounding nonminimality and a conjecture of Borovik-Cherlin
Bounding nonminimality and a conjecture of Borovik-Cherlin Open
Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The {\em degree o…
View article: Generic differential equations are strongly minimal
Generic differential equations are strongly minimal Open
In this manuscript we develop a new technique for showing that a nonlinear algebraic differential equation is strongly minimal based on the recently developed notion of the degree of nonminimality of Freitag and Moosa. Our techniques are s…
View article: Some functional transcendence results around the Schwarzian differential equation.
Some functional transcendence results around the Schwarzian differential equation. Open
This paper centers around proving variants of the Ax–Lindemann–Weierstrass (ALW) theorem for analytic functions which satisfy Schwarzian differential equations. In previous work, the authors proved the ALW theorem for the uniformizers of g…
View article: A differential approach to Ax-Schanuel, I
A differential approach to Ax-Schanuel, I Open
In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures; our general results describe the differential algebraic relations between the solutions of the partial differential equations satisfied by t…
View article: MODEL THEORY AND COMBINATORICS OF BANNED SEQUENCES
MODEL THEORY AND COMBINATORICS OF BANNED SEQUENCES Open
No description supplied
View article: Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups
Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups Open
We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the …
View article: Bounds in Query Learning
Bounds in Query Learning Open
We introduce new combinatorial quantities for concept classes, and prove lower and upper bounds for learning complexity in several models of query learning in terms of various combinatorial quantities. Our approach is flexible and powerful…
View article: MODEL THEORY AND MACHINE LEARNING
MODEL THEORY AND MACHINE LEARNING Open
About 25 years ago, it came to light that a single combinatorial property determines both an important dividing line in model theory (NIP) and machine learning (PAC-learnability). The following years saw a fruitful exchange of ideas betwee…
View article: Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian\n groups
Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian\n groups Open
We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the\nuniformizing functions of genus zero Fuchsian groups of the first kind. Our\nproof relies on differential Galois theory, monodromy of linear differential\nequations, t…
View article: Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups
Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups Open
We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the …
View article: Effective definability of Kolchin polynomials
Effective definability of Kolchin polynomials Open
While the natural model-theoretic ranks available in differentially closed fields (of characteristic zero), namely Lascar and Morley rank, are known not to be definable in families of differential varieties; in this note we show that the d…
View article: Strong minimality and the $j$-function
Strong minimality and the $j$-function Open
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 different…
View article: Algebraic relations between solutions of Painlevé equations
Algebraic relations between solutions of Painlevé equations Open
In this manuscript we make major progress classifying algebraic relations between solutions of Painlevé equations. Our main contribution is to establish the algebraic independence of solutions of various pairs of equations in the Painlevé …
View article: Bertini theorems for differential algebraic geometry
Bertini theorems for differential algebraic geometry Open
We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the different…