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Fundamental problems of algorithmic algebra
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Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Separating linear forms for bivariate systems
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Separating linear forms for bivariate systems
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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We address the problem of solving systems of two bivariate polynomials of total degree at most d 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 ÕB(d7+d6τ) and bound the bitsize of its coefficients by Õ(d2+dτ) (where ÕB 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 ÕB(d8+d7τ) 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 τ) at one real solution of the system in ÕB(d8+d7τ) bit operations and at all the ϴ(d2) solutions in only O(d) times that for one solution.