Book ChapterDOI
A computational method for diophantine approximation
Teresa Krick,Luis Miguel Pardo +1 more
- pp 193-253
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The procedures to solve algebraic geometry elimination problems have usually been designed from the point of view of commutative algebra as mentioned in this paper, which means that we have to eliminate a single block of quantifiers in a formula with polynomial equations.Abstract:
The procedures to solve algebraic geometry elimination problems have usually been designed from the point of view of commutative algebra. For instance, let us consider the problem of deciding whether a given system of polynomial equalities has a solution. This means that we have to eliminate a single block of quantifiers in a formula with polynomial equations.read more
Citations
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Journal ArticleDOI
Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)
TL;DR: In this article, the authors give a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients, and their study has developed into a fertile and extensive theory enriching many branches of pure mathematics.
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
A Gröbner Free Alternative for Polynomial System Solving
TL;DR: A new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials are introduced, and a new codification of the set of solutions of a positive dimensional algebraic variety is given relying on a new global version of Newton's iterator.
Journal ArticleDOI
Straight--Line Programs in Geometric Elimination Theory
TL;DR: A new method for solving symbolically zero-dimensional polynomial equation systems in the affine and toric case using Newton iteration in order to simplify straight-line programs occurring in elimination procedures and improving the well-know worst-case complexity bounds for zero- dimensional equation solving in symbolic and numeric computing.
Journal ArticleDOI
Sharp estimates for the arithmetic Nullstellensatz
TL;DR: In this article, the degree and height of polynomials in the integer ring ℤ over the integers have been derived for sparse polynomial systems, and the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field.
References
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Book
Commutative Ring Theory
Hideyuki Matsumura,Miles Reid +1 more
TL;DR: In this article, the authors introduce the notion of complete local rings and apply it to a wide range of applications, including: I-smoothness, I-flatness revisited, and valuation rings.
Book
Basic Algebraic Geometry
TL;DR: The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds as discussed by the authors, and is suitable for beginning graduate students.
Book
Fundamentals of Diophantine Geometry
TL;DR: In this article, the authors present a review of Lang's Diophantine Geometry, by L. J. Mordell and S. Lang, as well as a discussion of the relation between absolute values and proper sets of absolute values.
Book
Structural Complexity I
TL;DR: This volume is written for undergraduate students who have taken a first course in Formal Language Theory and presents the basic concepts of structural complexity, thus providing the background necessary for the understanding of complexity theory.
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.