Related papers: Solving Polynomial Equations from Complex Numbers
We give a general method for proving quantum lower bounds for problems with small range. Namely, we show that, for any symmetric problem defined on functions $f:\{1, ..., N\}\to\{1, ..., M\}$, its polynomial degree is the same for all…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
We explain how to obtain the set of solutions of a multivariate polynomial equation modulo a power of a prime number. These solutions are determined by a tree, called the trunk, which makes it possible to reconstruct all solutions. We apply…
We study the problem of reconstructing a function on a manifold satisfying some mild conditions, given data on the values and some derivatives of the function at arbitrary points on the manifold. While the problem of finding a polynomial of…
The necessary and sufficient conditions for a function to be totally or partially separable are derived. It is shown that a function is totally separable if and only if each component of the gradient vector of depends only on the…
We characterize the rational solutions to a KdV-like equation which are generated from polynomial solutions to the corresponding generalized bilinear equation. We use a particular class of polynomials satisfying a quadratic difference…
We consider an extremal problem for polynomials, which is dual to the well-known Smale mean value problem. We give a rough estimate depending only on the degree.
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
We consider the equation $P(Q(x_1,\ldots,x_\nu))=Q(P(x_1),\ldots,P(x_\nu))$ in polynomials over the field of complex numbers and prove that if ${\rm deg}(P)>1$, then it is only solvable in polynomials that are affinely conjugate to…
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.
This paper addresses the problem of deciding the lower-boundedness of an arbitrary real polynomial p in n variables.
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…
Criteria are given for determining whether an irreducible sextic equation with rational coefficients is algebraically solvable over the complex numbers.
Artin solved Hilbert's 17th problem, proving that a real polynomial in $n$ variables that is positive semidefinite is a sum of squares of rational functions, and Pfister showed that only $2^n$ squares are needed. In this paper, we…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
Student appreciation of a function is enhanced by understanding the graphical representation of that function. From the real graph of a polynomial, students can identify real-valued solutions to polynomial equations that correspond to the…
A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…