Related papers: Grothendieck-Katz conjecture

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed differential operator $\Lam \in k[\partial]$ follows from a special case of it,…

We show that an irreducible polynomial $p$ with no zeros on the closure of a matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that $p(0)=1$, admits a strictly contractive determinantal representation,…

In the paper, we find out the precise form of the finite order entire solutions of the following differential-difference equation \[f^{(k)}(z)=\sideset{}{^n_{j=0}}{\sum} a_j f(z+jc),\] where $a_0, a_1,\ldots,a_n(\neq 0)\in\mathbb{C}$. Also…

For a number field $k$ and an odd prime $p$, let $\tilde{k}$ be the compositum of all the ${\mathbb Z}_p$-extensions of $k$, $\tilde{\Lambda }$ the associated Iwasawa algebra, and $X(\tilde{k})$ the Galois group over $\tilde{k}$ of the…

We prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power-like zero order perturbations in bounded domains. The principal part of such equations is either…

We consider the model-theoretic Grothendieck ring of definable sets in ordered abelian groups. It is well-known that $\mathrm{K} \mathbb{Q} \cong \mathbb{Z}[T]/(T^2 + T)$ and $\mathrm{K} \mathbb{Z} =0$, but surprisingly little is known…

We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that…

We give a reformulation of the Lehmer conjecture about algebraic integers in terms of a simple counting problem modulo p.

We study the global structure of the set of radial solutions of a nonlinear Dirichlet problem involving the p-Laplacian with p>2, in the unit ball of $R^N$, $N \ges 1$. We show that all non-trivial radial solutions lie on smooth curves of…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

We consider the problem of uniqueness of positive solutions to boundary value problems containing the equation: -\Delta_p u =K(|x|)f(u), p>1. f is positive, is locally Lipschitz and satisfies some superlinear growth condition after u_0, a…

The product m_k of the first k primes (2..p_k) has neighbours m_k +/- 1 with all prime divisors beyond p_k, implying there are infinitely many primes [Euclid]. All primes between p_k and m_k are in the group G_1 of units in semigroup…

In this article we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of…

A special class of autonomous algebraic differential equations is studied. No equations in the class have any entire transcendental solutions. In a sense, for almost all equations in the class, transcendental meromorphic solutions can also…

We determine Grothendieck groups of periodic derived categories. In particular, we prove that the Grothendieck group of the $m$-periodic derived category of finitely generated modules over an Artin algebra is a free $\mathbb{Z}$-module if…

In this paper, we present some new results on the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Specifically, we show the (weak bi-anabelian and strong bi-anabelian) geometrically m-step solvable Grothendieck…

We prove that the Fermat-type equation $x^3 + y^3 = z^p$ has no solutions $(a,b,c)$ satisfying $abc \ne 0$ and $\gcd(a,b,c)=1$ when $-3$ is not a square mod~$p$. This improves to approximately $0.844$ the Dirichlet density of the set of…

Let $p$ and $q$ be locally H\"{o}lder functions in $\RR^N$, $p>0$ and $q\geq 0$. We study the Emden-Fowler equation $-\Delta u+ q(x)|\nabla u|^a=p(x)u^{-\gamma}$ in $\RR^N$, where $a$ and $\gamma$ are positive numbers. Our main result…

We consider the Paneitz-type equation $\Delta^2 u -\alpha \Delta u +\beta (u-u^q ) =0$ on a closed Riemannian manifold $(M,g)$. We reduce the equation to a fourth-order ordinary differential equation assuming that $(M,g)$ admits a proper…

In the present paper, we show a new result on the geometrically $2$-step solvable Grothendieck conjecture for genus $0$ curves over finitely generated fields. More precisely, we show that two genus $0$ hyperbolic curves over a finitely…