Given an algebra $A$ over a differential field $K$, we study derivations on $A$ that are compatible with the derivation on $K$. There is a universal object, which is a twisted version of the usual module of differentials, and we establish some of its basic properties. In the context of differential algebraic geometry, one gets a sheaf of these $τ$-differentials which can be interpreted as certain natural functions on the prolongation of a variety, as studied by Buium. This sheaf corresponds to the Kodaira-Spencer class of the variety.
We show that asymptotic (valued differential) fields have unique maximal immediate extensions. Connecting this to differential-henselianity, we prove that any differential-henselian asymptotic field is differential-algebraically maximal, removing the assumption of monotonicity from a theorem of Aschenbrenner, van den Dries, and van der Hoeven (arXiv:1509.02588, Theorem 7.0.3). Finally, we use this result to show the existence and uniqueness of differential-henselizations of asymptotic fields.
We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and use it to give a new class of examples of complete partial differential varieties. We use methods from both differential algebra and model theory.
We show that every valued differential field has an immediate strict extension that is spherically complete. We also discuss the issue of uniqueness up to isomorphism of such an extension.
So far there exist just a few results about the uniqueness of maximal immediate valued differential field extensions and about the relationship between differential-algebraic maximality and differential-henselianity; see arXiv:1509.02588, Chapter 7. We remove here the assumption of monotonicity in these results but replace it with the assumption that the value group is the union of its convex subgroups of finite (archimedean) rank. We also show the existence and uniqueness of differential-henselizations of asymptotic fields with such a value group.
The notion of newtonianity is central to the study of the ordered differential field of logarithmic-exponential transseries done by Aschenbrenner, van den Dries, and van der Hoeven; see Chapter 14 of arxiv:1509.02588. We remove the assumption of divisible value group from two of their results concerning newtonianity, namely the newtonization construction and the equivalence of newtonianity with asymptotic differential-algebraic maximality. We deduce the uniqueness of immediate differentially algebraic extensions that are asymptotically differential-algebraically maximal.
We develop a notion of (principal) differential rank for differential-valued fields, in analog of the exponential rank and of the difference rank. We give several characterizations of this rank. We then give a method to define a derivation on a field of generalized power series and use this method to show that any totally ordered set can be realized as the principal differential rank of a H-field.
In this paper we introduce a method of characteristic sets with respect to several term orderings for difference-differential polynomials. Using this technique, we obtain a method of computation of multivariate dimension polynomials of finitely generated difference-differential field extensions. Furthermore, we find new invariants of such extensions and show how the computation of multivariate difference-differential polynomials is applied to the equivalence problem for systems of algebraic difference-differential equations.
Kronecker's Theorem and Rabin's Theorem are fundamental results about computable fields F and the decidability of the set of irreducible polynomials over F. We adapt these theorems to the setting of differential fields K, with constrained pairs of differential polynomials over K assuming the role of the irreducible polynomials. We prove that two of the three basic aspects of Kronecker's Theorem remain true here, and that the reducibility in one direction (but not the other) from Rabin's Theorem also continues to hold.
Let $F$ be a field of characteristic $p$ and let $Ω^n(F)$ be the $F$-vector space of $n$-differential forms. In this work, we will study the annihilator of differential forms, give specific descriptions for special cases and show a connection between these annihilators and the kernels of the restriction map $Ω^n(F) \to Ω^n(E)$ for purely inseparable field extensions $E/F$.
In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic group (i.e. affine group scheme of finite type) over such a Laurent series field does occur as the differential Galois group of a linear differential equation with coefficients in any such function field (of one or several variables).
We define vector fields, leaves and trajectories for schemes. With these tools, we are able to give a geometrical interpretation and to generalize several results of differential Galois theory and constructions on differential schemes. We prove a theorem of extension of constant sections. Finally, as an application, we compare three classical sheaves defined over the differential spectrum.
An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions..
Let $G$ be one of the classical groups of Lie rank $l$. We make a similar construction of a general extension field in differential Galois theory for $G$ as E. Noether did in classical Galois theory for finite groups. More precisely, we build a differential field $E$ of differential transcendence degree $l$ over the constants on which the group $G$ acts and show that it is a Picard-Vessiot extension of the field of invariants $E^G$. The field $E^G$ is differentially generated by $l$ differential polynomials which are differentially algebraically independent over the constants. They are the coe…
We solve the inverse differential Galois problem over the fraction field of $k[[t,x]]$ and use this to solve split differential embedding problems over $k((t))(x)$ that are induced from $k(x)$. The proofs use patching as well as prior results on inverse problems and embedding problems.
Differential modules over a commutative differential ring R which are finitely generated projective as ring modules, with differential homomorphisms, form an additive category, so their isomorphism classes form a monoid. We study the quotient monoid of this monoid by the submonoid of isomorphism classes of free modules with component wise derivation. This quotient monoid has the reduced K group of R (ignoring the derivation) as an image and contains the reduced K group of the constants of R as its subgroup of units. This monoid provides a description of the isomorphism classes of differential …
Let $K$ be an inversive difference-differential field and $L$ a (not necessarily inversive) finitely generated difference-differential field extension of $K$. We consider the natural filtration of the extension $L/K$ associated with a finite system $η$ of its difference-differential generators and prove that for any intermediate difference-differential field $F$, the transcendence degrees of the components of the induced filtration of $F$ are expressed by a certain numerical polynomial $χ_{K, F,η}(t)$. This polynomial is closely connected with the dimension Hilbert-type polynomial of a submodu…
Let $F$ be a field of characteristic $p>0$ and let $Ω^n(F)$ be the $F$-vector space of $n$-differential forms over $F$. In this work we will study the behaviour of $Ω^n(F)$ under iterated function field extensions of $p$-forms. We will use results from a previous work to rewrite the kernel of the restriction map $Ω^n(F) \to Ω^n(F(φ_1,\ldots,φ_r))$ to a set of differential forms annihilated by specific forms given by the norm fields of $φ_1,\ldots,φ_r$. We will close this work by translating the new resluts to the theory of bilinear forms over fields of characteristic $2$.
The Fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations
Scuola Normale Superiore（高等师范学院）；CUNEF Universidad（CUNEF大学）
We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat. For the proof we develop a new characterization of free proalgebraic groups in terms of split embedding problems, and we use patching techniques in order to solve a very general class of differential embedding problems. Our result about $\mathbb{C}(x)$ also applies to rational function fields over more general fields of coefficients.
Let $C \langle \boldsymbol{t} \rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots, t_l)$ over an algebraically closed field $C$ of characteristic zero. In this article we present an explicit linear parameter differential equation over $C \langle \boldsymbol{t} \rangle$ with differential Galois group $\mathrm{SL}_{l+1}(C)$ and show that it is a generic equation in the following sense: If $F$ is an algebraically closed differential field with constants $C$ and $E/F$ is a Picard-Vessiot extension with differential Galois group $H(C) \subsete…
In this paper, we prove that every iterative differential embedding problem over an algebraic function field in positive characteristic with an algebraically closed field of constants has a proper solution.
