Related papers: On Sequences, Rational Functions and Decomposition

Noncommutative rational functions, i.e., elements of the universal skew field of fractions of a free algebra, can be defined through evaluations of noncommutative rational expressions on tuples of matrices. This interpretation extends their…

A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…

Graded modal logics generalise standard modal logics via families of modalities indexed by an algebraic structure whose operations mediate between the different modalities. The graded "of-course" modality $!_r$ captures how many times a…

In this paper we present algorithmic considerations and theoretical results about the relation between the orders of certain groups associated to the components of a polynomial and the order of the group that corresponds to the polynomial,…

We study a natural complexity measure of Boolean functions known as the rational degree. Denoted $\textrm{rdeg}(f)$, it is the minimal degree of a rational function that is equal to $f$ on the Boolean hypercube. For total functions $f$, it…

We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…

Although there are many simple proofs of Jordan's decomposition theorem in the literature (see [1], the references mentioned there, and [2]), our proof seems to be even more elementary. In fact, all we need is the theorem on the dimensions…

We present a package to perform partial fraction decompositions of multivariate rational functions. The algorithm allows to systematically avoid spurious denominator factors and is capable of producing unique results also when being applied…

Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…

Among all sequences that satisfy a divide-and-conquer recurrence, the sequences that are rational with respect to a numeration system are certainly the most immediate and most essential. Nevertheless, until recently they have not been…

We study the class of rational recursive sequences (ratrec) over the rational numbers. A ratrec sequence is defined via a system of sequences using mutually recursive equations of depth 1, where the next values are computed as rational…

We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…

This is a companion paper to arXiv:2312.10772. We deduce an equidistribution theorem for periodic nilsequences and use this theorem to give two applications in arithmetic combinatorics. The first application is quasi-polynomial bounds for a…

J.Ritt has investigated the structure of complex polynomials with respect to superposition. In particular, he listed all the polynomials admitting different double decompositions into indecomposable polynomials. The analogues of Ritt theory…

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This…

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words…

Many proofs of the fundamental theorem of algebra rely on the fact that the minimum of the modulus of a complex polynomial over the complex plane is attained at some complex number. The proof then follows by arguing the minimum value is…

Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general…

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly…