Differential Galois cohomology and parameterized Picard–Vessiot extensions

Assuming that the differential field [Formula: see text] is differentially large, in the sense of [León Sánchez and Tressl, Differentially large fields, preprint (2020); arXiv:2005.00888 ], and “bounded” as a field, we prove that for any linear differential algebraic group [Formula: see text] over [Formula: see text], the differential Galois (or constrained) cohomology set [Formula: see text] is finite. This applies, among other things, to closed ordered differential fields in the sense of [Singer, The model theory of ordered differential fields, J. Symb. Logic 43(1) (1978) 82–91], and to closed[Formula: see text]-adic differential fields in the sense of [Tressl, The uniform companion for large differential fields of characteristic [Formula: see text], Trans. Amer. Math. Soc. 357(10) (2005) 3933–3951]. As an application, we prove a general existence result for parameterized Picard–Vessiot (PPV) extensions within certain families of fields; if [Formula: see text] is a field with two commuting derivations, and [Formula: see text] is a parameterized linear differential equation over [Formula: see text], and [Formula: see text] is “differentially large” and [Formula: see text] is bounded, and [Formula: see text] is existentially closed in [Formula: see text], then there is a PPV extension [Formula: see text] of [Formula: see text] for the equation such that [Formula: see text] is existentially closed in [Formula: see text]. For instance, it follows that if the [Formula: see text]-constants of a formally real differential field [Formula: see text] is a closed ordered[Formula: see text]-field, then for any homogeneous linear [Formula: see text]-equation over [Formula: see text] there exists a PPV extension that is formally real. Similar observations apply to [Formula: see text]-adic fields.

Modifications of Gravity Via Differential Transformations of Field Variables

We discuss field theories appearing as a result of applying field transformations with derivatives (differential field transformations, DFTs) to a known theory. We begin with some simple examples of DFTs to see the basic properties of the procedure. In this process, the dynamics of the theory might either change or be conserved. After that, we concentrate on the theories of gravity which appear as a result of various DFTs applied to general relativity, namely the mimetic gravity and Regge–Teitelboim embedding theory. We review the main results related to the extension of dynamics in these theories, as well as the possibility to write down the action of a theory after DFTs as the action of the original theory before DFTs plus an additional term. Such a term usually contains some constraints with Lagrange multipliers and can be interpreted as an action of additional matter, which might be of use in cosmological applications, e.g., for the explanation of the effects of dark matter.

Nonlocal Gravitomagnetism

We briefly review the current status of nonlocal gravity (NLG), which is a classical nonlocalgeneralization of Einstein’s theory of gravitation based on a certain analogy with the nonlocalelectrodynamics of media. Nonlocal gravity thus involves integro-differential field equationsand a causal constitutive kernel that should ultimately be determined from observational data.We consider the stationary gravitational field of an isolated rotating astronomical source in the linearapproximation of nonlocal gravity. In this weak-field and slow-motion approximation of NLG,we describe the gravitomagnetic field associated with the rotating source and compare our resultswith gravitoelectromagnetism (GEM) of the standard general relativity theory. Moreover, we brieflystudy the energy-momentum content of the GEM field in nonlocal gravity.

Anharmonic solutions to the Riccati equation and elliptic modular functions

We study the irreducible algebraic equationx^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0,\quad\text{with ${n\geq 4}$,}on the differential field{(\mathbb{F}=\mathbb{C}(t),\delta=\frac{d}{dt})}. We assume that a root of the equation is a solution to the Riccati differential equation{u^{\prime}+B_{0}+B_{1}u+B_{2}u^{2}=0}, where{B_{0}},{B_{1}},{B_{2}}are in{\mathbb{F}}.We show how to construct a large class of polynomials as in the above algebraic equation, i.e., we prove that there exists a polynomial{F_{n}(x,y)\in\mathbb{C}(x)[y]}such that for almost{T\in\mathbb{F}\setminus\mathbb{C}}, the algebraic equation{F_{n}(x,T)=0}is of the same type as the above stated algebraic equation. In other words, all its roots are solutions to the same Riccati equation. On the other hand, we give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup{\Gamma(2)}, whose roots satisfy a common Riccati equation on the differential field{(\mathbb{C}(E_{2},E_{4},E_{6}),\frac{d}{d\tau})}, with{E_{i}(\tau)}being the Eisenstein series of weighti. These solutions are related to a Darboux–Halphen system. Finally, we deal with the following problem: For which “potential”{q\in\mathbb{C}(\wp,\wp^{\prime})}does the Riccati equation{\frac{dY}{dz}+Y^{2}=q}admit algebraic solutions over the differential field{\mathbb{C}(\wp,\wp^{\prime})}, with{\wp}being the classical Weierstrass function? We study this problem via Darboux polynomials and invariant theory and show that the minimal polynomial{\Phi(x)}of an algebraic solutionumust have a vanishing fourth transvectant{\tau_{4}(\Phi)(x)}.