Gröbner $δ$-bases and Gröbner bases for differential operators

Gr{ö}bner bases over Tate algebras

Tate algebras are fundamental objects in the context of analytic geometry over the p-adics. Roughly speaking, they play the same role as polynomial algebras play in classical algebraic geometry. In the present article, we develop the formalism of Gr{ö}bner bases for Tate algebras. We prove an analogue of the Buchberger criterion in our framework and design a Buchberger-like and a F4-like algorithm for computing Gr{ö}bner bases over Tate algebras. An implementation in SM is also discussed.

通用解析Gr{ö}bner基与Tropical几何

Universal Analytic Gr{ö}bner Bases and Tropical Geometry

本文提出一种通过遍历Gr{ö}bner扇计算多项式理想通用解析Gr{ö}bner基的算法，为计算tropical簇提供新视角，并探讨了凸多面体收敛约束下的局部基计算。

Geometric characterization of data sets with unique reduced Gröbner bases

Model selection based on experimental data is an important challenge in biological data science. Particularly when collecting data is expensive or time consuming, as it is often the case with clinical trial and biomolecular experiments, the problem of selecting information-rich data becomes crucial for creating relevant models. We identify geometric properties of input data that result in a unique algebraic model and we show that if the data form a staircase, or a so-called linear shift of a staircase, the ideal of the points has a unique reduced Gro bner basis and thus corresponds to a unique…

Logarithmic $p$-bases and arithmetical differential modules

We introduce the notion of log $p$-smoothness which weakens that of log-smoothness and that of having locally $p$-bases. We extend Berthelot's construction of arithmetic $D$-modules and some properties in this context.

Optimal bases for direct images of $p$-adic differential modules over discs

Let $(k,|\cdot|)$ be a complete and algebraically closed valued field extension of $(\mathbb{Q}_p,|\cdot|_p)$. Given a finite morphism $φ:\mathscr{D}_1\to \mathscr{D}_2$ of unit discs over $k$, a differential module $(M,D)$ on $\mathscr{D}_1$ and a point $b\in \mathscr{D}_2(k)$, we construct explicitly an optimal basis of space of horizontal elements for the direct image $φ_*(M,D)$ at $b$ in terms of the suitable chosen optimal bases of $(M,D)$ at preimages of $b$ by $φ$ and ramification properties of the morphism.

Equivariant quantum differential equation, Stokes bases, and K-theory for a projective space

We consider the equivariant quantum differential equation for the projective space $P^{n-1}$. We prove an equivariant gamma theorem for $P^{n-1}$, which describes the asymptotics of the differential equation at its regular singular point in terms of the equivariant characteristic gamma class of the tangent bundle of $P^{n-1}$. We describe the Stokes bases of the differential equation at its irregular singular point in terms of the exceptional bases of the equivariant K-theory algebra of $P^{n-1}$ and a suitable braid group action on the set of exceptional bases. Our results are an equivariant …