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Real Solving of Bivariate Polynomial Systems [chapter]

Ioannis Z. Emiris, Elias P. Tsigaridas
2005 Lecture Notes in Computer Science  
The methods employed are based on 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.  ... 
doi:10.1007/11555964_13 fatcat:3lolaooksnhtra7ykbbxwcmdvy

An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

Laureano González-Vega, M'hammed El Kahoui
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  ... 
doi:10.1006/jcom.1996.0032 fatcat:kw5h5qo6nfbfhkj6ibgm6n7n3u

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

Guadalupe Trujillo, Laureano González-vega
1997 Revista Matemática Complutense  
Tite main purpose of titis note is to show how 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.  ... 
doi:10.5209/rev_rema.1997.v10.17349 fatcat:hbqmuvnlmjavtpzyrq7tqsdtf4

Computations with one and two real algebraic numbers [article]

Ioannis Z. Emiris, Elias P. Tsigaridas
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] .  ... 
arXiv:cs/0512072v1 fatcat:liaima3txnfivml4hmke3c4gsi

Efficient topology determination of implicitly defined algebraic plane curves

Laureano Gonzalez-Vega, Ioana Necula
2002 Computer Aided Geometric Design  
Algebraic preliminaries: 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  ... 
doi:10.1016/s0167-8396(02)00167-x fatcat:jg32yugjlrcgnhngw2i2dphjgq

Fast and exact geometric analysis of real algebraic plane curves

Arno Eigenwillig, Michael Kerber, Nicola Wolpert
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 Sturm-Habicht sequence w.r.t. y.  ...  Sturm-Habicht sequences allow to count the number of distinct real roots in intervals (c, d).  ... 
doi:10.1145/1277548.1277570 dblp:conf/issac/EigenwilligKW07 fatcat:sleiz64kgbhn7hvs66axsa3zlm

Real algebraic numbers and polynomial systems of small degree

Ioannis Z. Emiris, Elias P. Tsigaridas
2008 Theoretical Computer Science  
Based on precomputed 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.  ... 
doi:10.1016/j.tcs.2008.09.009 fatcat:stoptrld3vfh3jlzv2xaninqry

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

Silvia Gandy, Masaaki Kanno, Hirokazu Anai, Kazuhiro Yokoyama
2011 Mathematics in Computer Science  
A discussion of the connection of the 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.  ... 
doi:10.1007/s11786-011-0090-5 fatcat:ivihxxxhizcwfgv6rlptymkqde

SyNRAC: A Maple-Package for Solving Real Algebraic Constraints [chapter]

Hirokazu Anai, Hitoshi Yanami
2003 Lecture Notes in Computer Science  
,v+1 is called the 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] .  ... 
doi:10.1007/3-540-44860-8_86 fatcat:wzicqmszxvci3kvse2s367hrmu

Direct symbolic transformation from 3D cartesian into hyperboloidal coordinates

Gema María Díaz-Toca, Ioana Necula
2014 Applied Mathematics and Computation  
The analysis of the polynomial's roots is performed by an algebraically complete stratification, based on symbolic techniques (mainly 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 ).  ... 
doi:10.1016/j.amc.2013.11.099 fatcat:wlbivjxvvjgcngpvgrisdrhifq

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.  ... 

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.  ... 

An Algorithm for Computing the Complete Root Classification of a Parametric Polynomial [chapter]

Songxin Liang, David J. Jeffrey
2006 Lecture Notes in Computer Science  
It is shown that the discriminant 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.  ... 
doi:10.1007/11856290_12 fatcat:qxd7o7di6fdpddvixtnfjeltcq

Comparing Real Algebraic Numbers of Small Degree [chapter]

Ioannis Z. Emiris, Elias P. Tsigaridas
2004 Lecture Notes in Computer Science  
We show a new method to compare real algebraic numbers by precomputing generalized 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.  ... 
doi:10.1007/978-3-540-30140-0_58 fatcat:usjj3k4rkrb4nk7l3m6jq4et4q
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