Improved algorithm for computing separating linear forms for bivariate systems
Identifieur interne : 002993 ( Hal/Corpus ); précédent : 002992; suivant : 002994Improved algorithm for computing separating linear forms for bivariate systems
Auteurs : Yacine Bouzidi ; Sylvain Lazard ; Guillaume Moroz ; Marc Pouget ; Fabrice RouillierSource :
Abstract
We address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and this is the bottleneck of these algorithms in terms of worst-case bit complexity. We present for this problem a new algorithm of worst-case bit complexity $\sOB(d^7+d^6\tau)$ where $d$ and $\tau$ denote respectively the maximum degree and bitsize of the input (and where $\sO$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). This algorithm simplifies and decreases by a factor $d$ the worst-case bit complexity presented for this problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields, for this problem, a probabilistic Las-Vegas algorithm of expected bit complexity $\sOB(d^5+d^4\tau)$.
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<sourceDesc><biblStruct><analytic><title xml:lang="en">Improved algorithm for computing separating linear forms for bivariate systems</title>
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<front><div type="abstract" xml:lang="en">We address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and this is the bottleneck of these algorithms in terms of worst-case bit complexity. We present for this problem a new algorithm of worst-case bit complexity $\sOB(d^7+d^6\tau)$ where $d$ and $\tau$ denote respectively the maximum degree and bitsize of the input (and where $\sO$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). This algorithm simplifies and decreases by a factor $d$ the worst-case bit complexity presented for this problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields, for this problem, a probabilistic Las-Vegas algorithm of expected bit complexity $\sOB(d^5+d^4\tau)$.</div>
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<hal api="V3"><titleStmt><title xml:lang="en">Improved algorithm for computing separating linear forms for bivariate systems</title>
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<sourceDesc><biblStruct><analytic><title xml:lang="en">Improved algorithm for computing separating linear forms for bivariate systems</title>
<author role="aut"><persName><forename type="first">Yacine</forename>
<surname>Bouzidi</surname>
</persName>
<email>yacine.bouzidi@inria.fr</email>
<idno type="halAuthorId">595230</idno>
<affiliation ref="#struct-205129"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Sylvain</forename>
<surname>Lazard</surname>
</persName>
<email>sylvain.lazard@inria.fr</email>
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<author role="aut"><persName><forename type="first">Guillaume</forename>
<surname>Moroz</surname>
</persName>
<email>guillaume.moroz@inria.fr</email>
<idno type="halAuthorId">537926</idno>
<affiliation ref="#struct-205129"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Marc</forename>
<surname>Pouget</surname>
</persName>
<email>marc.pouget@inria.fr</email>
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<idno type="halAuthorId">1023487</idno>
<idno type="arXiv">http://arxiv.org/a/Pouget_M</idno>
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</author>
<author role="aut"><persName><forename type="first">Fabrice</forename>
<surname>Rouillier</surname>
</persName>
<email>Fabrice.Rouillier@inria.fr</email>
<idno type="idHal">fabrice-rouillier</idno>
<idno type="halAuthorId">74826</idno>
<idno type="arXiv">http://arxiv.org/a/fabricerouillier</idno>
<affiliation ref="#struct-22"></affiliation>
<affiliation ref="#struct-206119"></affiliation>
</author>
</analytic>
<monogr><meeting><title>ISSAC - 39th International Symposium on Symbolic and Algebraic Computation</title>
<date type="start">2014-07-23</date>
<date type="end">2014-07-25</date>
<settlement>Kobe</settlement>
<country key="JP">Japan</country>
</meeting>
<imprint><date type="datePub">2014-07</date>
</imprint>
</monogr>
<idno type="arxiv">1405.4740</idno>
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<abstract xml:lang="en">We address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and this is the bottleneck of these algorithms in terms of worst-case bit complexity. We present for this problem a new algorithm of worst-case bit complexity $\sOB(d^7+d^6\tau)$ where $d$ and $\tau$ denote respectively the maximum degree and bitsize of the input (and where $\sO$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). This algorithm simplifies and decreases by a factor $d$ the worst-case bit complexity presented for this problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields, for this problem, a probabilistic Las-Vegas algorithm of expected bit complexity $\sOB(d^5+d^4\tau)$.</abstract>
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