On a solution functor for D-cap-modules via pp-adic Hodge theory

This article aims to formulate the main technical input for the construction of a solution functor in a still hypothetical $p$-adic analytic Riemann-Hilbert correspondence. Our approach relies on a novel period sheaf $\mathbb{B}_{\text{la}}^{+}$, which is a certain ind-Banach completion of $\mathbb{B}_{\text{inf}}$ along the kernel of Fontaine's map $\theta$. We relate the ind-continuous $\mathbb{B}_{\text{la}}^{+}$-linear endomorphisms of a corresponding period structure sheaf to the sheaf of infinite order differential operators D-cap introduced by Ardakov-Wadsley. Locally on the cotangent bundle, this yields a definition of a solution functor. The main result computes that significant information about a perfect complex can be reconstructed out of its solutions, which hints strongly towards a reconstruction theorem for a large category of complexes of D-cap-modules

No short polynomials vanish on bounded rank matrices

We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$-minors or $t$-Pfaffians of a generic matrix or skew-symmetric matrix one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a sufficiently general $t$-dimensional subspace of an affine $n$-space does not contain polynomials with fewer than $t+1$ terms.Comment: 13 pages, comments welcome, v2: 15 pages, final version as in Bulletin LM