Book ChapterDOI
Some Useful Bounds
M. Mignotte
- pp 259-263
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TLDR
In this paper, the determinant of a matrix, the absolute value of the roots of a polynomial, the coefficients of divisors of polynomials, and the minimal distance between the root vectors are discussed.Abstract:
Some fundamental inequalities for the following values are listed: the determinant of a matrix, the absolute value of the roots of a polynomial, the coefficients of divisors of polynomials, and the minimal distance between the roots of a polynomial. These inequalities are useful for the analysis of algorithms in various areas of computer algebra.read more
Citations
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Proceedings ArticleDOI
Some algebraic and geometric computations in PSPACE
TL;DR: A PSPACE algorithm for determining the signs of multivariate polynomials at the common zeros of a system of polynomial equations is given and it is shown that the existential theory of the real numbers can be decided in PSPACE.
Journal ArticleDOI
On the combinatorial and algebraic complexity of quantifier elimination
TL;DR: This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data, and new and improved algorithms for deciding a sentence in the first order theory over real closed fields, are obtained.
Journal ArticleDOI
The Geometry of the Zeros of a Polynomial in a complex variable. By M. Marden. Pp. ix, 183. $5. 1949. Mathematical Surveys, 3. (American Mathematical Society)
Proceedings ArticleDOI
A new algebraic method for robot motion planning and real geometry
TL;DR: An algorithm which solves the findpath or generalized movers' problem in single exponential sequential time, the first algorithm for the problem whose sequential time bound is less than double exponential and an algebraic tool called the multivariate resultant which gives a necessary and sufficient condition for a system of homogeneous polynomials to have a solution.
Journal ArticleDOI
Hilbert's Nullstellensatz Is in the Polynomial Hierarchy
TL;DR: A new lower bound for unsatisfiable systems is included, and remarks on the Arthur-Merlin class are included, on the problem of deciding whether a system of polynomial equations in several complex variables has a solution.
References
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Journal ArticleDOI
The Geometry of the Zeros of a Polynomial in a complex variable. By M. Marden. Pp. ix, 183. $5. 1949. Mathematical Surveys, 3. (American Mathematical Society)
Proceedings ArticleDOI
Some inequalities about univariate polynomials
TL;DR: This paper deals with the two following topics: bounds for the heights of divisors of polynomials, minimal distance between distinct roots of integral univariate polynomsials, and new inequalities in the best known results.