Related papers: Additive combinatorics with a view towards compute…
Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject as well as external…
This is a survey of old and new problems and results in additive number theory.
This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is…
When inclusions in a composite are separated by a very small gap, high contrast between the inclusion and matrix properties can induce strong amplification of the underlying field inside the narrow region. Quantifying this field…
This article is a survey on the topic of polynomial amoebas. We review results of papers written on the topic with an emphasis on its computational aspects. Polynomial amoebas have numerous applications in various domains of mathematics and…
Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we…
The combination of the group ring setting with the methods of character theory allows an elegant and powerful analysis of various combinatorial structures, via their character sums. These combinatorial structures include difference sets,…
The seminal concept of characteristic polygon of an embedded algebroid surface, first developed by Hironaka, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object…
New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> makes an important such contribution.
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
This is an exposition of the contributions of L\'aszl\'o Lov\'asz to mathematics and computer science written on the occasion of the bestowal of the Abel Prize~2021 to him. Our survey, of course, cannot be exhaustive. We sketch remarkable…
The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and…
Combinatorial Game Theory is a branch of mathematics and theoretical computer science that studies sequential 2-player games with perfect information. Normal play is the convention where a player who cannot move loses. Here, we generalize…
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has…
We introduce new types of local algorithms, which we call "ASI Algorithms", and use them to demonstrate a link between descriptive and computable combinatorics. This allows us to unify arguments from the two fields, and also sometimes to…
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of…
Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial…
This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…
We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. As all previously known examples, our examples are contracted semigroup algebras and the underlying…