Related papers: Projective spaces over $\mathbb{F}_{1^{\ell}}$
We study projective functions. We prove that projective functions generalise lower and upper-semianalytic ones while being stable by composition and difference. We show that the class of projective functions is closed under sums,…
Let A be a commutative Noetherian ring of dimension d and let P be a projective R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac {1}{f_1\ldots f_m}]-module of rank r\geq max {2,dim A+1, where f_i\in A[Y_i]. Then (i) \EL^1(R\op P) acts transitively…
We compare three notions of effectiveness on uncountable structures. The first notion is that of a $\real$-computable structure, based on a model of computation proposed by Blum, Shub, and Smale, which uses full-precision real arithmetic.…
Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior)…
We study finite $0$-dimensional schemes in product of multiprojective spaces and their ideals. In particular, we describe the set of generators of the ideal defining a $0$-dimensional scheme in the case $\mathbb P^{1}\times\cdots…
We introduce a family of mathematical objects called $\mathcal{P}$-schemes, where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A $\mathcal{P}$-scheme is a collection of partitions of the right coset spaces $H\backslash G$,…
A projection space is a collection of spaces interrelated by the combinatorics of projection onto tensor factors in a symmetric monoidal background category. Examples include classical configuration spaces, orbit configuration spaces, the…
A structural analysis of construction schemes is developed. That analysis is used to give simple and new constructions of combinatorial objects which have been of interest to set theorists and topologists. We then continue the study of…
Regions-based theories of space aim -- among others -- to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed,…
Starting from the guiding principles of spacetime locality and operationalism, a general framework for a probabilistic description of nature is proposed. Crucially, no notion of time or metric is assumed, neither any specific physical…
Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes,…
This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta…
This text serves as an introduction to $\mathbb{F}_1$-geometry for the general mathematician. We explain the initial motivations for $\mathbb{F}_1$-geometry in detail, provide an overview of the different approaches to $\mathbb{F}_1$ and…
This paper is devoted to the open problem in $\mathbb{F}_1$-geometry of developing $K$-theory for $\mathbb{F}_1$-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for…
We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for the hypothetical "curve" $C=\overline {\Sp \Z}$ over $\F_1$, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of…
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
We refine the notion of variety over the "field with one element" developed by C. Soul\'e by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint…
We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point. After introducing…
First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective…
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for…