Related papers: Monomials in arithmetic circuits: Complete problem…
A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by…
The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its…
We study the problem of generating monomials of a polynomial in the context of enumeration complexity. In this setting, the complexity measure is the delay between two solutions and the total time. We present two new algorithms for…
In this paper we obtain complexity bounds for computational problems on algebraic power series over several commuting variables. The power series are specified by systems of polynomial equations: a formalism closely related to weighted…
In~\cite{algorithmic} was given an algorithm that computes arithmetical structures on matrices. We use some of the ideas contained there to get an algorithm that computes arithmetical structures over dominated polynomials. A dominated…
The numerical construction of polynomials in the product representation (as used for instance in variants of the multiboson technique) can become problematic if rounding errors induce an imprecise or even unstable evaluation of the…
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its…
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates.…
Binary polynomial optimization is equivalent to the problem of minimizing a linear function over the intersection of the multilinear set with a polyhedron. Many families of valid inequalities for the multilinear set are available in the…
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
There are many extremely challenging problems about existence of monochromatic arithmetic progressions in colorings of groups. Many theorems hold only for abelian groups as results on non-abelian groups are often much more difficult to…
The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common…
We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…
We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is…
We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Gr\"obner bases, factorization or sub-resultant computations.
In this paper, we devise two algorithms for the problem of testing $q$-monomials of degree $k$ in any multivariate polynomial represented by a circuit, regardless of the primality of $q$. One is an $O^*(2^k)$ time randomized algorithm. The…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…