Related papers: New Method for Constructing Complete Cap Sets
In this paper, an upper bound for the number of integral points of bounded height on an affine complete intersection defined over $\mathbb{Z}$ is proven. The proof uses an extension to complete intersections of the method used for…
This article provides a new perspective on the geometry of a projective line, which helps clarify and illuminate some classical results about projective plane. As part of the same train of ideas, the article also provides a proof of the…
We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the 'multiplicative' property of filtrations on the corresponding completions and…
The affine Hilbert function is a classical algebraic object that has been central, among other tools, to the development of the polynomial method in combinatorics. Owing to its concrete connections with Gr\"obner basis theory, as well as…
A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. Necessary and sufficient conditions for the reconstructibility of…
There are two properties shared by all known crossing-minimizing geometric drawings of $K_n$, for $n$ a multiple of 3. First, the underlying $n$-point set of these drawings has exactly $3\binom{k+2}{2}$ $(\le k)$-edges, for all $0\le k <…
Unsupervised point cloud completion aims at estimating the corresponding complete point cloud of a partial point cloud in an unpaired manner. It is a crucial but challenging problem since there is no paired partial-complete supervision that…
For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over $k=\mathbb{F}_{p^2}$, that is optimal if $p=3$.
In this paper, we first discussed multiplicative metric mapping by giving some topological properties of the relevant multiplicative metric space. As an interesting result of our discussions, we observed that the set of positive real…
We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e., projective systems. It is known that the maximum number of $(h-1)$-spaces in PG$(2,q)$, such that no hyperplane…
Point cloud completion aims to reconstruct complete 3D shapes from partial observations, which is a challenging problem due to severe occlusions and missing geometry. Despite recent advances in multimodal techniques that leverage…
We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices…
We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a…
This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the…
Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm…
Given positive integers $n,k$ with $k\leq n$, we consider the number of ways of choosing $k$ subsets of $\{1,\ldots,n\}$ in such a way that the union of these subsets gives $\{1,\ldots,n\}$ and they are not subsets of each other. We refer…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
Erd\H{o}s asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? F\"uredi constructed a set of $n$…
Tree sets are posets with additional structure that generalize tree-like objects in graphs, matroids, or other combinatorial structures. They are a special class of abstract separation systems. We study infinite tree sets and how they…
Hadwiger's covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an…