Separating linear forms and Rational Univariate Representations of bivariate systems

Abstract

International audienceWe 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 worst-case bit complexity. Given two bivariate polynomials of total degree at most $d$ with integer coefficients of bitsize at most~$\tau$, our algorithm computes a separating linear form {of bitsize $O(\log d)$} in \comp\ bit operations in the worst case, which decreases by a factor $d^2$ the best known complexity for this problem (where $\sO$ 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 worst-case bit complexity $\sOB(d^7+d^6\tau)$ and that its coefficients have bitsize $\sO(d^2+d\tau)$.} 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 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 $\tau$) at one real solution of the system in $\sOB(d^{8}+d^7\tau)$ bit operations and at all the $\Theta(d^2)$ real solutions in only $O(d)$ times that for one solution

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

“…, where m as in (8). Consider the primal-dual multiplication maps (see paragraph 4.1) δ r ij : Vij → Wr , for all i, j ∈ {0,1,2}, i < j and r ∈ {i, j}, expressing multiplication by fs where {s} = {i, j} \ {r}.…”

Improved algorithm for computing separating linear forms for bivariate systems

“…It is straightforward that, in Line 1, the sheared polynomials P (t − sy, y) and Q(t − sy, y) can be computed in bit complexity OB (d 4 + d 3 τ ) and that their bitsizes are in O(d + τ ) (see e.g. [5,Lemma 7]). In Lines 2 and 3, the polynomials, in one or two variables, have degree at most d and bitsize O(d + τ ).…”

“…Note that this algorithm performs O(d 4 ) arithmetic operations in F (see e.g. [5,Lemma 15]). We also state the following properties which directly follow from the algorithm and Lemma 2.…”

“…In Algorithm 5, we compute a lucky prime for {H, ∂H ∂y } (see Definition 5) in a straightforward manner by first computing the number of distinct solutions of the system and then by computing the number of solutions of its image modulo distinct prime numbers µ until the same number of solutions is found (and checking that some leading coefficients do not vanish modulo µ). Note that Algorithm 5 is a simplified variant of [5,Algorithm 3] where we use here the knowledge of the number of critical points of H to avoid computing an explicit bound on the number of unlucky primes. Proof.…”

“…More recently, Bouzidi et al [5] presented a new algorithm for computing a separating linear form that reduces the previous bit complexity to OB(d 8 + d 7 τ ). The same authors also showed that, given such a separating linear form, an alternative rational parameterization called RUR [14] can be computed using OB (d 7 + d 6 τ ) bit operations [5,Thm. 22] and that isolating boxes of the solutions can be computed from this RUR in OB (d 6 + d 5 τ ) [4, Thm.…”