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Real Solving of Bivariate Polynomial Systems
[chapter]

2005
*
Lecture Notes in Computer Science
*

The methods employed are based on

doi:10.1007/11555964_13
fatcat:3lolaooksnhtra7ykbbxwcmdvy
*Sturm*-*Habicht**sequences*, univariate resultants and rational univariate representation. ... We use cascaded*Sturm*-*Habicht**sequences*. ... Let P denote the*Sturm*-*Habicht**sequence*of P and P . For a*Sturm*-*Habicht**sequence*P , V P (p) denotes the number of sign variations of the evaluation of the*sequence*at p. ...##
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An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

1996
*
Journal of Complexity
*

*Sturm*-

*Habicht*

*Sequence*Let ބ be an ordered integral domain. This section is devoted to introducing the definition of the

*Sturm*-

*Habicht*

*sequence*and its main properties. ...

*STURM*-

*HABICHT*

*SEQUENCE*AND REAL ALGEBRAIC NUMBERS This section is devoted to introducing the properties of the

*Sturm*-

*Habicht*

*sequence*and Thom's coding of real algebraic numbers, which are going to be ...

##
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Multivariate Sturm-Habicht sequences:realroot counting on n-rectangles and triangles

1997
*
Revista Matemática Complutense
*

Tite main purpose of titis note is to show how

doi:10.5209/rev_rema.1997.v10.17349
fatcat:hbqmuvnlmjavtpzyrq7tqsdtf4
*Sturm*-*Habicht**Sequence*can be generalized to the multivariate case and used to compute tbe number of real solutions of a polynamial system of equations with ...*Sturm*-*Habicht**Sequence*is ane of tite toals titat Computational Real Algebraic Geometry provides to deal witit tite prablem of computing tite number of real roots of an univariate polynomial in 7Z[x] witit ...*Sturm*-*Habicht**Sequence*Let 1< be an ordered fleld and F a real-closed fleid with 11< C F. ...##
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Computations with one and two real algebraic numbers
[article]

2005
*
arXiv
*
pre-print

*sequences*. ... complexity results concerning computations with one and two real algebraic numbers, as well as real solving of univariate polynomials and bivariate polynomial systems with integer coefficients using

*Sturm*-

*Habicht*... For the

*Sturm*-

*Habicht*(or Sylvester-

*Habicht*)

*sequences*the reader may refer to the work of Gonzalez-Vega et al [21, 22] . ...

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Efficient topology determination of implicitly defined algebraic plane curves

2002
*
Computer Aided Geometric Design
*

Algebraic preliminaries:

doi:10.1016/s0167-8396(02)00167-x
fatcat:jg32yugjlrcgnhngw2i2dphjgq
*Sturm*-*Habicht**sequence*This section is devoted to introduce the definition of the*Sturm*-*Habicht**sequence*and its main properties related with the real root counting problem. ... Next the*Sturm*-*Habicht**sequence*is displayed together with the principal*Sturm*-*Habicht*coefficients: H 4 (x, y) = StHa 4 (P ) := y 4 − 6x + 4x 2 y 2 + 24x 3 + x 2 ; h 4 (x) = 1. ... The process is restarted after applying the change of variable The new polynomial f (x − y, y) will be denoted again by f (x, y): The*Sturm*-*Habicht**sequence*together with the principal*Sturm*-*Habicht*coefficients ...##
###
Fast and exact geometric analysis of real algebraic plane curves

2007
*
Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07
*

For a polynomial f ∈ D[x, y], let (StHai(f )) n i=0 be its

doi:10.1145/1277548.1277570
dblp:conf/issac/EigenwilligKW07
fatcat:sleiz64kgbhn7hvs66axsa3zlm
*Sturm*-*Habicht**sequence*w.r.t. y. ...*Sturm*-*Habicht**sequences*allow to count the number of distinct real roots in intervals (c, d). ...##
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Real algebraic numbers and polynomial systems of small degree

2008
*
Theoretical Computer Science
*

Based on precomputed

doi:10.1016/j.tcs.2008.09.009
fatcat:stoptrld3vfh3jlzv2xaninqry
*Sturm*-*Habicht**sequences*, discriminants and invariants, we classify, isolate with rational points, and compare the real roots of polynomials of degree up to 4. ... In our approach, we evaluate only one*Sturm*-*Habicht**sequence*, over two rational numbers. ...*Sturm**sequences*and real algebraic numbers*Sturm*(and*Sturm*-*Habicht*), e.g. [1, 13, 15, 22] ,*sequences*is a well known and useful tool for isolating the roots of any polynomial. ...##
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Page 5235 of Mathematical Reviews Vol. , Issue 99h
[page]

1999
*
Mathematical Reviews
*

The article under review deals with a generalization to the mul- tivariate case of the

*Sturm*-*Habicht**sequence*. ... This multivariate*Sturm*-*Habicht**sequence*is used to compute the number of real solutions of a polynomial system of equations with a finite num- ber of complex solutions. ...##
###
Optimizing a Particular Real Root of a Polynomial by a Special Cylindrical Algebraic Decomposition

2011
*
Mathematics in Computer Science
*

A discussion of the connection of the

doi:10.1007/s11786-011-0090-5
fatcat:ivihxxxhizcwfgv6rlptymkqde
*Sturm*-*Habicht**sequence*and real root counting can be found in the Appendix. ... Our prototype program outperformed the conventional approach that employs conditions from the*Sturm*-*Habicht**sequence*. ...##
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SyNRAC: A Maple-Package for Solving Real Algebraic Constraints
[chapter]

2003
*
Lecture Notes in Computer Science
*

,v+1 is called the

doi:10.1007/3-540-44860-8_86
fatcat:wzicqmszxvci3kvse2s367hrmu
*Sturm*-*Habicht**sequence*of P . We simply denote it by {SH j (P )}. The*Sturm*-*Habicht**sequence*can be used for real root counting in almost the same way as the*Sturm**sequence*. ... Special QE Using the*Sturm*-*Habicht**Sequence*A special QE method based on the*Sturm*-*Habicht**sequence*for the first-order formula ∀x f(x) > 0, where f (x) ∈ R[x] was proposed in [4] . ...##
###
Direct symbolic transformation from 3D cartesian into hyperboloidal coordinates

2014
*
Applied Mathematics and Computation
*

The analysis of the polynomial's roots is performed by an algebraically complete stratification, based on symbolic techniques (mainly

doi:10.1016/j.amc.2013.11.099
fatcat:wlbivjxvvjgcngpvgrisdrhifq
*Sturm*-*Habicht**sequences*and its properties related to real root counting ... The*Sturm*-*Habicht**sequence*associated to P and Q is defined as the polynomial list {StHa j (P, Q)} j=0,... ... In the particular case of Q = 1, the*Sturm*-*Habicht**sequence*associated to P are denoted by StHa j (P ), and the*Sturm*-*Habicht*principal coefficients associated to P by stha j (P ). ...##
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Page 3203 of Mathematical Reviews Vol. , Issue 95f
[page]

1995
*
Mathematical Reviews
*

They show the link between these methods and the so-called

*Sturm*-*Habicht**sequence*, and the link between this*sequence*and the subresultant theory. ... The second part of the paper contains a detailed study of the*Sturm*-*Habicht**sequence*. The authors describe and compare several methods for counting the number of real roots of a polynomial. ...##
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Page 7672 of Mathematical Reviews Vol. , Issue 2000k
[page]

2000
*
Mathematical Reviews
*

[in
12 FIELD THEORY AND POLYNOMIALS
7672
Quantifier elimination and cylindrical algebraic decomposition (Linz, 1993), 300-316, Springer, Vienna, 1998; see MR 99b:03007] intro- duced the

*Sturm*-*Habicht**sequence*... The*sequence*obtained contains no zero-polynomials and no revision of signs is needed; it is clear that our*sequence*is a proper*Sturm**sequence*. ...##
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An Algorithm for Computing the Complete Root Classification of a Parametric Polynomial
[chapter]

2006
*
Lecture Notes in Computer Science
*

It is shown that the discriminant

doi:10.1007/11856290_12
fatcat:qxd7o7di6fdpddvixtnfjeltcq
*sequences*, upon which the sign lists are based, are closely related both to*Sturm*-*Habicht**sequences*and to subresultant*sequences*. ... The improvement lies in the direct use of 'sign lists', obtained from the discriminant*sequence*, rather than 'revised sign lists'. ... p , and the*Sturm*-*Habicht**sequence*of p and 1 are all the same. ...##
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Comparing Real Algebraic Numbers of Small Degree
[chapter]

2004
*
Lecture Notes in Computer Science
*

We show a new method to compare real algebraic numbers by precomputing generalized

doi:10.1007/978-3-540-30140-0_58
fatcat:usjj3k4rkrb4nk7l3m6jq4et4q
*Sturm**sequences*, thus avoiding iterative methods; the method, moreover handles all degenerate cases. ... Our second contribution is to exploit invariants and Bezoutian subexpressions in writing the*sequences*, in order to reduce bit complexity. ... The*Sturm*-*Habicht**sequence*was computed in maple, using the elements of the Bezoutian matrix. ...
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