# Improved algorithm for computing separating linear forms for bivariate systems

1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
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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Conference papers
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Cited literature [15 references]

https://hal.inria.fr/hal-00992634
Contributor : Marc Pouget <>
Submitted on : Monday, May 19, 2014 - 10:08:38 AM
Last modification on : Friday, April 10, 2020 - 5:23:05 PM
Long-term archiving on: : Monday, April 10, 2017 - 11:32:12 PM

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### Identifiers

• HAL Id : hal-00992634, version 1
• ARXIV : 1405.4740

### Citation

Yacine Bouzidi, Sylvain Lazard, Guillaume Moroz, Marc Pouget, Fabrice Rouillier. Improved algorithm for computing separating linear forms for bivariate systems. ISSAC - 39th International Symposium on Symbolic and Algebraic Computation, Jul 2014, Kobe, Japan. ⟨hal-00992634⟩

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