We provide a characterization of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to characterize differential largeness in terms of being existentially closed in the differential algebraic Laurent series ring, and we prove that any large field of infinite transcendence degree can be expanded to a differentially large field even under certain prescribed constant fields. As an application, we show that the theory of proper dense pairs of models of a complete and model-complete theory of large fields, is a complete theory. As a further consequence of the expansion result we show that there is no real closed and differential field that has a prime model extension in closed ordered differential fields, unless it is itself a closed ordered differential field.

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 primitive permutation group is definably isomorphic to an algebraic permutation group living in the constants. Applications include the verification, in differentially closed fields, of the finite Morley rank permutation group conjectures of Borovik-Deloro and Borovik-Cherlin. Applying the results to binding groups for internality to the constants, it is deduced that if complete types p and q are of rank m and n, respectively, and are nonorthogonal, then the $(m+3)$rd Morley power of p is not weakly orthogonal to the $(n+3)$rd Morley power of q. An application to transcendence of generic solutions of pairs of algebraic differential equations is given.

Let ${\mathbb K}$ be an algebraically bounded structure, and let T be its theory. If T is model complete, then the theory of ${\mathbb K}$ endowed with a derivation, denoted by $T^{\delta }$, has a model completion. Additionally, we prove that if the theory T is stable/NIP then the model completion of $T^{\delta }$ is also stable/NIP. Similar results hold for the theory with several derivations, either commuting or non-commuting.

We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007).

We generalize Buium’s notion of an algebraic D-group to ${\mathcal {L}}$-definable D-groups, namely $(G,s)$, where G is an ${\mathcal {L}}$-definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$-definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form

$$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$

for some ${\mathcal {L}}$-definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$).

We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic $0$.

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property is first order.

We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation.

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 sufficient to show that generic order $h$ differential equations with non-constant coefficients are strongly minimal, answering a question of Poizat (1980).

This paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $\alpha$, respectively, we solve entire solutions of the Tumura–Clunie type differential equation $f^{n}+P(z,\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $n\geq 2$ is an integer, $P(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a nontrivial solution such that $\lambda (f)<\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\not =0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l> s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$. In particular, we precisely characterize all solutions such that $\lambda (f)<\infty$ when $l=2$ and $l=4$.

A multivariate, formal power series over a field K is a Bézivin series if all of its coefficients can be expressed as a sum of at most r elements from a finitely generated subgroup $G \le K^*$; it is a Pólya series if one can take $r=1$. We give explicit structural descriptions of D-finite Bézivin series and D-finite Pólya series over fields of characteristic $0$, thus extending classical results of Pólya and Bézivin to the multivariate setting.

We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p, the p-primary part of A is either finite or it coincides with the Prüfer p-group. We also provide a model-theoretic description of the model companions we obtain.

Every discrete definable subset of a closed asymptotic couple with ordered scalar field ${\boldsymbol {k}}$ is shown to be contained in a finite-dimensional ${\boldsymbol {k}}$-linear subspace of that couple. It follows that the differential-valued field $\mathbb {T}$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $\mathbb {R}$ as the underlying set for the second sort.

We develop tools for constructing rigid analytic trivializations for Drinfeld modules as infinite products of Frobenius twists of matrices, from which we recover the rigid analytic trivialization given by Pellarin in terms of Anderson generating functions. One advantage is that these infinite products can be obtained from only a finite amount of initial calculation, and consequently we obtain new formulas for periods and quasi-periods, similar to the product expansion of the Carlitz period. We further link to results of Gekeler and Maurischat on the $\infty $ -adic field generated by the period lattice.

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators.

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 size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including, for example, difference fields).

We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for $p$-adic differential equations $Dx=0$ on the $p$-adic open unit disc $|t|<1$, which measure the asymptotic behavior of solutions $x$ as $|t|\to 1^{-}$. Then, Dwork calculated the log-growth filtration for $p$-adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for $(\varphi ,\nabla )$-modules over $K[\![t]\!]_0$, which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to $(\varphi ,\nabla )$-modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$-varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme $Z$ of finite type to an algebraic variety $X$, with the property that there are infinitely many hypersurfaces on $X$ whose scheme-theoretic inverse images under $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$. In the case where $Z$ is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic ${\mathcal{D}}$-varieties and of Cantat’s theorem to self-correspondences.

Let P and Q be relatively prime integers greater than 1, and let f be a real valued discretely supported function on a finite dimensional real vector space V. We prove that if $f_{P}(x)=f(Px)-f(x)$ and $f_{Q}(x)=f(Qx)-f(x)$ are both $\Lambda $ -periodic for some lattice $\Lambda \subset V$ , then so is f (up to a modification at $0$ ). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.

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 universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:

A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields and of the theory of separably closed fields of arbitrary imperfection degree.

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over $\mathbb{Q}$. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over $\mathbb{Q}_{p}(x)$.