Related papers: On The Local Sum Conjecture in Two Dimensions

In this paper, we show local Gersten's conjecture for regular system of parameters. As its consequence we obtain Gersten's conjecture for a commutative regular local ring and smooth over a commutative discrete valuation ring.

In this paper we present a new approach to prove effective results in Diophantine approximation. We then use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation with…

Any counterexample to the two-dimensional Jacobian Conjecture gives a rational map from one projective plane to another. We use some ideas of the Minimal Model Program to study the combinatorial structure of a rational surface, that is…

We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension.

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$ and $\widehat{M}$ its convex hull. We say that $M$ is locally polynomially integrable if the $(n-1)-$ dimensional volumes of the sections of $\widehat M$ by…

In this article we introduce generalized projective spaces (Definitions $[2.1, 2.5]$) and prove three main theorems in two different contexts. In the first context we prove, in main Theorem $A$, the surjectivity of the Chinese remainder…

Let $J(m)$ be an $m\times m$ Jordan block with eigenvalue $1$. For $\lambda\in \mathbb{C}\setminus\{0,1\}$, we explicitly construct all rank $2$ local systems of geometric origin on $\mathbb{P}^1\setminus\{0,1,\lambda, \infty\}$, with local…

This paper establishes an analogue of the special chain theorem for the embedding dimension of polynomial rings, with direct application on the (embedding) codimension. In particular, we recover a classic result on the transfer of…

The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the…

We show that the Volume Conjecture for polyhedra implies a weak version of the Stoker Conjecture; in turn we prove that this weak version of the Stoker conjecture implies the Stoker conjecture. The main tool used is an extension of a result…

For finite sets A and B in the plane, we write A+B to denote the set of sums of the elements of A and B. In addition, we write tr(A) to denote the common number of triangles in any triangulation of the convex hull of A using the points of A…

This paper gives an algorithm to determine whether a number in a biquadratic field is a sum of two squares, based on local-global principle of isotropy of quadratic forms.

In this note, we establish the validity of a conjecture recently proposed in Mathematics Magazine and connect it to the existing interesting results

Given two parties performing experiments in separate laboratories, we provide a diagrammatic formulation of what it means for the joint statistics of their experiments to satisfy local realism. In particular, we show that the principles of…

This article is part introduction and part survey to the mathematical area centered around local cohomology.

We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…

We prove the rank one case of Skolem's Conjecture on the exponential local-global principle for algebraic functions and discuss its analog for meromorphic functions.

We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored Jones polynomials.

We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.

In this paper, we prove a structure theorem for the infinite union of $n$-adic doubling measures via techniques which involve far numbers. Our approach extends the results of Wu in 1998, and as a by product, we also prove a classification…