On the theory of resolvents and its applications
01 Jan 1999-
TL;DR: A generalized algorithm to construct the generalized resolvents based on Wu-Rits's zero decomposition algorithm and a complete method of finding parametricequations for algebraic curves are presented.
Abstract: We extend the concept of the resolvent of a prime ideal to the concept of theresolvent of a general ideal with respect to a set of parameters and propose an algorithmto construct the generalized resolvents based on Wu-Rits's zero decomposition algorithm.Our generalized algorithm has the following applications. (1) For a reducible variety V,we can find a direction on which V is projected birationally to an irreducible hypersurface.(2) We give a new algorithm to find a primitive element for a finite algebraic extensionof a field of characteristic zero. (3) We present a complete method of finding parametricequations for algebraic curves. (4) We give a method of solving a system of polynomialequations to any given precision.
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TL;DR: This paper reports a geometric constraint-solving approach based on symbolic computation that can compute robust numerical solutions for a set of equations and give complete methods of deciding whether the constraints are independent and whether a constraint system is over-constraint.
Abstract: This paper reports a geometric constraint-solving approach based on symbolic computation. With this approach, we can compute robust numerical solutions for a set of equations and give complete methods of deciding whether the constraints are independent and whether a constraint system is over-constraint. Based on symbolic computation, we also have a decision procedure for the problem of deciding whether a constrained diagram can be constructed with ruler and compass (rc-constructibility).
93 citations
28 Jul 2009
TL;DR: A local generic position method is proposed to isolate the real roots of a bivariate polynomial system using the multiplicities of the roots of Σ=0 are the same as that of the corresponding roots of T(X)=0.
Abstract: A local generic position method is proposed to isolate the real roots of a bivariate polynomial system ∑={f(x,y),g(x,y)}. In this method, the roots of the system are represented as linear combinations of the roots of two univariate polynomial equations t(x)=0 and T(X)=0: {x = α, y = β -- α/s | α e V(t(x)), β e V(T(X)), ||β -- α|
43 citations
Cites methods from "On the theory of resolvents and its..."
...The concept of generic position was used in equation solving and topology determination for a long time [1, 4, 5, 7, 10 , 12, 13, 15, 17, 21]....
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TL;DR: It is shown that the generic zeros of a differential ideal A defined by a differential chain A are birationally equivalent to the general zerosof a single regular differential polynomial, providing a generalization of both the cyclic vector construction for system of linear differential equations and the rational univariate representation of algebraic zero dimensional radical ideals.
Abstract: We show that the generic zeros of a differential ideal [A]:H
∞
A
defined by a differential chain A are birationally equivalent to the general zeros of a single regular differential polynomial. This provides a generalization of both the cyclic vector construction for system of linear differential equations and the rational univariate representation of algebraic zero dimensional radical ideals. In order to achieve generality, we prove new results on differential dimension and relative orders which are of independent interest.
36 citations
TL;DR: A root isolating algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation, which can give the exact precisions needed for isolating the roots of the univariate equations.
27 citations
Cites background from "On the theory of resolvents and its..."
...One of the basic methods to solve polynomial equation systems is based on the concept of separating elements, which can be traced back to Kronecker [14] and has been studied extensively in the past twenty years [1, 2, 4, 8, 9, 10, 11, 12, 13, 15, 19, 20, 24]....
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TL;DR: In this paper, the authors improved the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials.
19 citations
References
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01 Jan 1950
TL;DR: A linear transformation with rational maps riemann sphere is presented in this article, where the presentation is kept as elementary as A linear transformations with rational map Riemann spheres the converse is where sense.
Abstract: This introduction to algebraic geometry examines how the more recent abstract concepts relate to traditional analytical and geometrical problems. The presentation is kept as elementary as A linear transformations with rational maps riemann sphere the converse is where sense. Any quantity positive or division process. Any combination of such groups and an expression or four xi. The category of points has fixed points.
1,365 citations
"On the theory of resolvents and its..." refers background in this paper
...A rational curve always has a set of proper parametric equations [21]....
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...By Bezout’s theorem [21], the degree of f = 0 equals the number of the intersection points between f = 0 and a generic straight line....
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01 Jan 1985
1,322 citations
Additional excerpts
...y1 2 ])....
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...we ask what accuracy for x1 is needed if we want a certain accuracy for xp. In [ 2 ], this is considered to be an inherent di‐culty of polynomial equation solving....
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01 Jan 1975
TL;DR: In this paper, a quantifier elimination method for the elementary theory of real closed fields is presented. But it does not provide a decision method, which enables one to decide whether any sentence of the theory is true or false, since many important and difficult mathematical problems can be expressed in this theory.
Abstract: Tarski in 1948, (Tarski 1951) published a quantifier elimination method for the elementary theory of real closed fields (which he had discovered in 1930). As noted by Tarski, any quantifier elimination method for this theory also provides a decision method, which enables one to decide whether any sentence of the theory is true or false. Since many important and difficult mathematical problems can be expressed in this theory, any computationally feasible quantifier elimination algorithm would be of utmost significance.
951 citations
TL;DR: This paper presents the theory of well-ordering of polynomials and a constructive theory of algebraic varieties based on the work of J. F. Ritt, and presents the detailed proofs of the basic principles underlying this method.
Abstract: At the end of 1976 and the beginning of 1977, the author discovered a mechanical method for proving theorems in elementary geometries. This method can be applied to various unordered elementary geometries satisfying the Pascalian Axiom, or to theorems not involving the concept of ‘order’ (e.g., thatc is ‘between’a andb) in various elementary geometries. In Section 4 we give the detailed proofs of the basic principles underlying this method. In Sections 2 and 3 we present the theory of well-ordering of polynomials and a constructive theory of algebraic varieties. Our method is based on these theories, both of which are based on the work of J. F. Ritt. In Section 5 we use Morley's theorem and the Pascal-conic theorem discovered by the author to illustrate the computer implementation of the method.
393 citations
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01 Jan 1983
TL;DR: 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.
160 citations
"On the theory of resolvents and its..." refers background in this paper
...Since R(w0) = 0, ‖w0‖ < C = 1 + M/m [18]....
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...362, [18]), we know that a lower bound for the distances among the roots of Vi is...
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