Rational univariate representations of bivariate systems and applications

We address the problem of solving systems of two bivariate polynomials of total degree at most <i>d</i> with integer coefficients of maximum bitsize τ We suppose known a linear separating form (that is a linear combination of the variables that takes different values at distinct solutions of the system) and focus on the computation of a Rational Univariate Representation (RUR). We present an algorithm for computing a RUR with worst-case bit complexity in Õ<sub>B</sub>(d<sup>7</sup>+d<sup>6</sup>τ) and bound the bitsize of its coefficients by Õ(d<sup>2</sup>+dτ) (where Õ<sub>B</sub> refers to bit complexities and Õ to complexities where polylogarithmic factors are omitted). We show in addition that isolating boxes of the solutions of the system can be computed from the RUR with Õ<sub>B</sub>(d<sup>8</sup>+d<sup>7</sup>τ) bit operations. Finally, we show how a RUR can be used to evaluate the sign of a bivariate polynomial (of degree at most <i>d</i> and bitsize at most τ) at one real solution of the system in Õ<sub>B</sub>(d<sup>8</sup>+d<sup>7</sup>τ) bit operations and at all the ϴ(d<sup>2</sup>) solutions in only <i>O</i>(<i>d</i>) times that for one solution.

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