Rational Univariate Representations of Bivariate Systems and Applications

Yacine Bouzidi (1) , Sylvain Lazard (1) , Marc Pouget (1) , Fabrice Rouillier (2)

1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
2 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs

We address the problem of solving systems of two bivariate polynomials of total degree at most $d$ with integer coefficients of maximum bitsize $\tau$. 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 $\sOB(d^7+d^6\tau)$ and bound the bitsize of its coefficients by $\sO(d^2+d\tau)$ (where $O_B$ refers to bit complexities and $\sO$ 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 $\sOB(d^{8}+d^7\tau)$ bit operations. 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 $\tau$) at one real solution of the system in $\sOB(d^{8}+d^7\tau)$ bit operations and at all the $\Theta(d^2)$ solutions in only $O(d)$ times that for one solution.

Mots clés: Bivariate system; Rational univariate representation

Domaines: Géométrie algorithmique [cs.CG]

Yacine Bouzidi, Sylvain Lazard, Marc Pouget, Fabrice Rouillier. Rational Univariate Representations of Bivariate Systems and Applications. ISSAC - 38th International Symposium on Symbolic and Algebraic Computation, Jun 2013, Boston, United States. pp.109-116. ⟨hal-00809430⟩