Related papers: Multiple polylogarithms, cyclotomy and modular com…
There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm. We introduce an Euler complex data on modular…
Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function $Li_(z)$. The…
Short review article on quantum computation accepted for Supplement III, Encyclopaedia of Mathematics (publication expected Summer 2001). See also http://www.wkap.nl/series.htm/ENM
This article has been withdrawn because it has been merged with the earlier article GCT3 (arXiv: CS/0501076 [cs.CC]) in the series. The merged article is now available as: Geometric Complexity Theory III: on deciding nonvanishing of a…
We improve several recent results by Hong, Lee, Lee and Park (2012) on gaps and Bzd\c{e}ga (2014) on jumps amongst the coefficients of cyclotomic polynomials. Besides direct improvements, we also introduce several new techniques that have…
This is a review article on mirror symmetry and aspects of it related to the theory of modular forms. We describe this topic along its historical development and connect to some more recent results toward the end. The article is for…
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
In this talk I review the connections between Feynman integrals and multiple polylogarithms. After an introductory section on loop integrals I discuss the Mellin-Barnes transformation and shuffle algebras. In a subsequent section multiple…
In "On the homotopy theory of arrangements," published in 1986, the authors gave a comprehensive survey of the subject. This article updates and continues the earlier article, noting some key open problems.
In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of the multiple polylogarithms and degenerate versions of the multi-poly-Bernoulli numbers and polynomials. We investigate…
This paper presents a closed form polynomial expression for the binary cyclotomic polynomial. We contrast this against expressions for binary cyclotomic polynomials in (Lam and Leung 1996) and (Lenstra 1979).
This text consists of five relatively systematic notes on Gr\"obner bases and free resolutions of modules over solvable polynomial algebras.
This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
General revision. In particular the parts concerning involutive bases over rings have been significantly changed. In addition some proofs have been improved.
A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods…
These are course notes I wrote for my Fall 2013 graduate topics course on geometric structures, taught at ICERM. The notes rework many of proofs in William P. Thurston's beautiful but hard-to-understand paper, "Shapes of Polyhedra". A…
The paper is written for Kluwer's Encyclopaedia of Mathematics.
This is an English translation of the following paper, published several years ago: Nikonorov Yu.G., Nikonorova Yu.V. Generalized Popoviciu's problem (Russian), Tr. Rubtsovsk. Ind. Inst., 7, 229-232 (2000), Zbl. 0958.51021. All inserted…