Related papers: An Algorithmic Characterization of Polynomial Func…

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

Vladimir Shpilrain and Jie-Tai Yu have asked for an effective algorithm to decide if two elements of C[x,y] are related by an automorphism of C[x,y]. We describe here an efficient algorithm that decides this question and finds the…

A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to…

Polynomial networks and factorization machines are two recently-proposed models that can efficiently use feature interactions in classification and regression tasks. In this paper, we revisit both models from a unified perspective. Based on…

In this paper we turn the spotlight on a class of lexicographic ranking functions introduced by Bradley, Manna and Sipma in a seminal CAV 2005 paper, and establish for the first time the complexity of some problems involving the inference…

Can we effectively learn a nonlinear representation in time comparable to linear learning? We describe a new algorithm that explicitly and adaptively expands higher-order interaction features over base linear representations. The algorithm…

When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…

We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for, parametric optimization. The idea is to first consider the variable x1 as a parameter and solve the…

Let E_n={x_i=1, x_i+x_j=x_k, x_i*x_j=x_k: i,j,k \in {1,...,n}}. We prove: (1) there is an algorithm that for every computable function f:N-->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any…

The sequential allocation protocol is a simple and popular mechanism to allocate indivisible goods, in which the agents take turns to pick the items according to a predefined sequence. While this protocol is not strategy-proof, it has been…

The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by $A \in\mathbb{Z}^{m\times{}n}$ and present an algorithm to solve such problems in polynomial-time provided that both the…

A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables…

We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its…

Motivated by algorithmic problems from combinatorial group theory we study computational properties of integers equipped with binary operations +, -, z = x 2^y, z = x 2^{-y} (the former two are partial) and predicates < and =. Notice that…

Many combinatorial optimisation problems can be modelled as valued constraint satisfaction problems. In this paper, we present a polynomial-time algorithm solving the valued constraint satisfaction problem for a fixed number of variables…

We consider the problem of the computation of $\inf_p \theta p$ over the set of exponent pairs $P \ni p$ under linear constraints for a certain class of objective functions $\theta$. An effective algorithm is presented. The output of the…

The valence of a function f at a point $z_0$ is the number of distinct, finite solutions to $f(z) = z_0.$ In this paper, we bound the valence of complex-valued harmonic polynomials in the plane for some special harmonic polynomials of the…

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is…

Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while…