Set Of Multivariate Polynomials Research Articles

Clustering in the Lazard method for Cylindrical Algebraic Decomposition

Cylindrical Algebraic Decomposition (CAD) is an algorithm originally proposed by Collins in [4]. Given a set of multivariate polynomials, CAD decomposes the multidimensional real space into connected subsets called cells, within which those polynomials are sign-invariant.

Tight Polynomial Bounds for Loop Programs in Polynomial Space

We consider the following problem: given a program, find tight asymptotic bounds on the values of some variables at the end of the computation (or at any given program point) in terms of its input values. We focus on the case of polynomially-bounded variables, and on a weak programming language for which we have recently shown that tight bounds for polynomially-bounded variables are computable. These bounds are sets of multivariate polynomials. While their computability has been settled, the complexity of this program-analysis problem remained open. In this paper, we show the problem to be PSPACE-complete. The main contribution is a new, space-efficient analysis algorithm. This algorithm is obtained in a few steps. First, we develop an algorithm for univariate bounds, a sub-problem which is already PSPACE-hard. Then, a decision procedure for multivariate bounds is achieved by reducing this problem to the univariate case; this reduction is orthogonal to the solution of the univariate problem and uses observations on the geometry of a set of vectors that represent multivariate bounds. Finally, we transform the univariate-bound algorithm to produce multivariate bounds.

On arithmetic-geometric-mean polynomials

ABSTRACTWe study here an aspect of an infinite set of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of for which every element may be expressed as a finite sum of squares of real polynomials. Some of the mathematics discussed here would be suitable as extension material for senior pre-university students and first-year undergraduates.

Algebraic reasoning for the enhancement of data-driven building reconstructions

Data-driven approaches for the reconstruction of buildings feature the flexibility needed to capture objects of arbitrary shape. To recognize man-made structures, geometric relations such as orthogonality or parallelism have to be detected. These constraints are typically formulated as sets of multivariate polynomials. For the enforcement of the constraints within an adjustment process, a set of independent and consistent geometric constraints has to be determined. Gröbner bases are an ideal tool to identify such sets exactly. A complete workflow for geometric reasoning is presented to obtain boundary representations of solids based on given point clouds. The constraints are formulated in homogeneous coordinates, which results in simple polynomials suitable for the successful derivation of Gröbner bases for algebraic reasoning. Strategies for the reduction of the algebraical complexity are presented. To enforce the constraints, an adjustment model is introduced, which is able to cope with homogeneous coordinates along with their singular covariance matrices. The feasibility and the potential of the approach are demonstrated by the analysis of a real data set.

Algebraic independence in positive characteristic: A $p$-adic calculus

A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic p > 0, no analogous characterization is known. In this paper we give the first such criterion. Essentially, it boils down to a non-degeneracy condition on a lift of the Jacobian polynomial over (an unramified extension of) the ring of p-adic integers. Our proof builds on the functorial de Rham-Witt complex, which was invented by Illusie (1979) for crystalline cohomology computations, and we deduce a natural explicit generalization of the Jacobian. We call this new avatar the Witt-Jacobian. In essence, we show how to faithfully differentiate polynomials over Fp (i.e., somehow avoid ∂xp/∂x = 0) and thus capture algebraic independence. We give two applications of this criterion in algebraic complexity theory.

Reduced stability testing set for a diamond-type family of quasipolynomials

A diamond-type family of quasipolynomials, for which vertex stability results hold, is presented. Both delay independent and delay-dependent stability conditions are given. In the first case, it suffices to check for the stability of a finite testing set of multivariate polynomials. While in the other one, it is also needed to check the stability of several edge sub-families of quasipolynomials.

A C# package for assembling quantum circuits and generating associated polynomial sets

Recently it has been shown that elements of the unitary matrix determined by a quantum circuit can be computed by counting the number of common roots in the finite field ℤ2 for a certain set of multivariate polynomials over ℤ2. In this paper we present a C# package that allows a user to assemble a quantum circuit and to generate the multivariate polynomial set associated with the circuit. The generated polynomial system can further be converted to the canonical triangular involutive basis form, which is appropriate for counting the number of common roots of the polynomials.

Stability of Multivariate Polynomials, Part 4: Conic Sets

Robust stability of conic sets of multivariate polynomials is the main topic of the paper. Stability and semistability conditions for a cone starting at the origin are obtained along with stability conditions for a cone of the form p( s) + K, where p( s) is a given polynomial and K is a cone starting at the origin. Some new results on convex directions for mulivariate polynomials are also presented.

Accurate enclosure of the zero set of multivariate polynomials

The zero set of one general multivariate polynomial is enclosed by unions and intersections of funnel-shaped unbounded sets. There are sharper enclosures for the zero set of a polynomial in two complex variables with complex interval coefficients. Common zeros of a polynomial system can be located by an appropriate intersection of these enclosure sets in an appropriate space. The resulting domain is directly brought into polynomial equation solvers.

An Elimination Method for Polynomial Systems

We present an elimination method for polynomial systems, in the form of three main algorithms. For any given system [P,Q] of two sets of multivariate polynomials, one of the algorithms computes a sequence of triangular forms T1,…,Te and polynomial sets U1,…,Ue such that Zero(P/Q) = ∪ei=1 Zero(Ti/Ui), where Zero(P/Q) denotes the set of common zeros of the polynomials in P which are not zeros of any polynomial in Q, and similarly for Zero(Ti/Ui). The two other algorithms compute the same zero decomposition but with nicer properties such as Zero(Ti/Ui) ≠ &0slash; for each i. One of them, for which the computed triangular systems [Ti, Ui] possess the projection property, provides a quantifier elimination procedure for algebraically closed fields. For the other, the computed triangular forms Ti are irreducible. The relationship between our method and some existing elimination methods is explained. Experimental data for a set of test examples by a draft implementation of the method are provided, and show that the efficiency of our method is comparable with that of some well-known methods. A few encouraging examples are given in detail for illustration.