Solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers

Classification of p-adic 6-dimensional filiform Leibniz algebras by solution of x^q=a

In this paper we study the $p$-adic equation $x^q=a$ over the field of $p$-adic numbers. We construct an algorithm of calculation of criteria of solvability in the case of $q=p^m$ and present a computer program to compute the criteria for fixed value of $m \leq p-1$. Moreover, using this solvability criteria for $q=2,3,4,5,6$, we classify $p$-adic 6-dimensional filiform Leibniz algebras.

Roth's solvability criteria for the matrix equations ${AX-\widehat XB=C}$ and ${X-A\widehat{X}B=C}$ over the skew field of quaternions with an involutive automorphism $q\mapsto \hat q$

The matrix equation $AX-XB=C$ has a solution if and only if the matrices [A&C\\0&B] and [A &0\\0 & B] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) obtained an analogous criterion for the matrix equation $X-AXB=C$ over a field. We extend these criteria to the matrix equations $AX-\widehat XB=C$ and $X-A\widehat XB=C$ over the skew field of quaternions with a fixed involutive automorphism $q\mapsto \hat q$.

广义塞维斯特型方程的可解性和唯一性准则

Solvability and uniqueness criteria for generalized Sylvester-type equations

本文研究了广义$\star$-塞维斯特矩阵方程$AXB + CX^\star D = E$的可解性和唯一性条件，给出了一种保证对于任意右端E都存在唯一解的充分必要条件，并扩展了现有结果到任意系数矩阵的情况。

Completed tensor products and a global approach to $p$-adic analytic differential operators

Ardakov-Wadsley defined the sheaf D-cap of $p$-adic analytic differential operators on a smooth rigid analytic variety $X$ by restricting to the case where $X$ is affinoid and the tangent sheaf admits a smooth Lie lattice. We generalize their results by dropping the assumption of a smooth Lie lattice throughout, which allows us to describe the sections of D-cap for arbitrary affinoid subdomains and not just on a suitable base of the topology. The structural results concerning D-cap and coadmissible D-cap-modules can then be generalized in a natural way. The main ingredient for our proofs is a …