Related papers: Solving parametric radical equations with depth 2 …
The real radical ideal of a system of polynomials with finitely many complex roots is generated by a system of real polynomials having only real roots and free of multiplicities. It is a central object in computational real algebraic…
Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…
Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the…
We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a polynomial $f\in \mathbb{Z}[x_1, \ldots, x_k]$ and nonnegative integers $a_1, \ldots, a_k$ and $d_1, \ldots,$ $d_k$, written in binary, test…
The paper deals with the resolution of third and fourth degree equations by means of radicals. It is a survey of some historical details about this fundamental problem. Moreover, it explains practical methods for the resolution of third and…
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 study the problem of identifying the parameters of a linear system from its response to multiple unknown waveforms. We assume that the system response is a scaled superposition of time-delayed and frequency-shifted versions of the…
We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.
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…
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…
We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets A \subset Z and show quantitative bounds on the size of A if there are no non-trivial solutions. We use the circle method…
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in…
Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In this…
Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a…
Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for…
This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via…
First-order systems of hyperbolic partial differential equations (PDEs) occur ubiquitously throughout computational physics, commonly used in simulations of fluid turbulence, shock waves, electromagnetic interactions, and even general…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
A major open problem in proof complexity is to demonstrate that random 3-CNFs with a linear number of clauses require super-polynomial size refutations in bounded-depth Frege systems. We take the first step towards addressing this question…
The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild…