Solving bivariate systems using Rational Univariate Representations
Résumé
Given two coprime polynomials $P$ and $Q$ in $\Z[x,y]$ of degree bounded by
$d$ and bitsize bounded by $\tau$, we address the problem of solving the
system $\{P,Q\}$. We are interested in certified numerical approximations or,
more precisely, isolating boxes of the solutions.
We are also interested in computing, as intermediate symbolic objects, rational parameterizations of
the solutions, and in particular Rational Univariate Representations (RURs), which can easily turn many queries on the system into queries on univariate polynomials.
Such representations
require the computation of a separating
form for the system, that is a linear combination of the variables that takes
different values when evaluated at the distinct solutions of the system.
We present new algorithms for computing linear separating forms, RUR
decompositions and isolating boxes of the solutions. We show that these three algorithms
have worst-case bit complexity $\widetilde{O}_B(d^6+d^5\tau)$, where $\widetilde{O}$ refers to the
complexity where polylogarithmic factors are omitted and $O_B$ refers to the
bit complexity. We also present probabilistic Las Vegas variants of our two
first algorithms, which have expected bit complexity $\widetilde{O}_B(d^5+d^4\tau)$.
A key ingredient of our proofs of complexity
is an amortized analysis of the triangular
decomposition algorithm via subresultants, which is of independent interest.
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