Related papers: $p$-adic Integral Geometry

This is a brief review article of various applications of non-Archimedean geometry, p-adic numbers and adeles in modern mathematical physics.

Let $C$ be a genus $2$ curve with Jacobian isomorphic to the square of an elliptic curve with complex multiplication by a maximal order in an imaginary quadratic field of discriminant $-d<0$. We show that if the stable model of $C$ has bad…

In this paper, we extend the previous convergence results for the generalized alternating projection method applied to subspaces in [arXiv:1703.10547] to hold also for smooth manifolds. We show that the algorithm locally behaves similarly…

We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of…

We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and the weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining…

This paper is a sequel to arXiv:2501.14444, in which we shall give proofs of several results stated in arXiv:2501.14444 (Theorems D--L) which, for brevity and clarity, we postponed to this sequel paper. These results were the following: for…

This paper first introduces the concept of p-adic number and field. Then it develops the p-adic integration and applied it to solve p-adic Schrodinger equations.

In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…

We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work…

We explicitly construct simple, piecewise minimizing geodesic, arbitrarily fine interpolation of simple and Jordan curves on a Riemannian manifold. In particular, a finite sequence of partition points can be specified in advance to be…

In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.

Vologodsky's theory of $p$-adic integration plays a central role in computing several interesting invariants in arithmetic geometry. In contrast with the theory developed by Coleman, it has the advantage of being insensitive to the…

Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…

We attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational…

Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…

We use Stein's method to obtain bounds on the rate of convergence for a class of statistics in geometric probability obtained as a sum of contributions from Poisson points which are exponentially stabilizing, i.e. locally determined in a…

For $G$ a symplectic or orthogonal $p$-adic group (not necessarily split), or an inner form of a general linear $p$-adic group, we compute the endomorphism algebras of some induced projective generators \`a la Bernstein of the category of…

In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.

We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.

In this paper we prove that the motivic Eisenstein classes associated to polylogarithms of commutative group schemes can be $p$-adically interpolated in \'etale cohomology. This generalizes results for elliptic curves obtained in our former…