Details:
Title  Separating linear forms and Rational Univariate Representations of bivariate systems  Author(s)  Yacine Bouzidi, Sylvain Lazard, Marc Pouget, Fabrice Rouillier  Type  Article in Journal  Abstract  Abstract We address the problem of solving systems of bivariate polynomials with integer coefficients. We first present an algorithm for computing a separating linear form of such systems, that is a linear combination of the variables that takes different values when evaluated at distinct (complex) solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. 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, moreover, the computation of a separating linear form is the bottleneck of these algorithms, in terms of worstcase bit complexity. Given two bivariate polynomials of total degree at most d with integer coefficients of bitsize at most τ, our algorithm computes a separating linear form of bitsize O(log ⁡d) in O ˜ B(d^8 + d^7 τ) bit operations in the worst case, which decreases by a factor d 2 the best known complexity for this problem (where O ˜ refers to the complexity where polylogarithmic factors are omitted and O B refers to the bit complexity). We then present simple polynomial formulas for the Rational Univariate Representations (RURs) of such systems. This yields that, given a separating linear form of bitsize O(log ⁡ d) , the corresponding RUR can be computed in worstcase bit complexity O ˜ B(d^7 + d^6 τ) and that its coefficients have bitsize O ˜ (d^2 + dτ) . We show in addition that isolating boxes of the solutions of the system can be computed from the RUR with O ˜ B(d^8 + d^7 τ) bit operations in the worst case. Finally, we show how a RUR can be used to evaluate the sign of a bivariate polynomial (of degree at most d and bitsize at most τ) at one real solution of the system in O ˜ B(d^8 + d^7 τ) bit operations and at all the Θ(d^2) real solutions in only O(d) times that for one solution.  Keywords  Bivariate system, Separating linear form, Rational univariate representation  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717114000595 
Language  English  Journal  Journal of Symbolic Computation  Volume  68, Part 1  Number  0  Pages  84  119  Year  2015  Edition  0  Translation 
No  Refereed 
No 
