Separating linear forms for bivariate systems

Abstract

We present an algorithm for computing a separating linear form of a system of bivariate polynomials with integer coefficients, 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 τ , our algorithm computes a separating linear form in e OB(d2 ) bit operations in the worst case, where the previously known best bit complexity for this problem was e OB(d(where e O refers to the complexity where polylogarithmic factors are omitted and OB refers to the bit complexity).

Rational univariate representations of bivariate systems and applications

“…Moreover, such a separating form, with a < 2d 4 , can be computed in e OB(d 8 +d 7 τ +d 5 τ 2 ) bit operations [4]. As a direct consequence of Propositions 6 and 10, we get the following result.…”

“…It is thus sufficient to analyze the complexity of computing, for every α, a box JX,α × JY,α that contains α and such that the widths of these intervals are smaller than half of 2 −ε . 4 For technical reasons, we require that the interval widths are smaller than 2 −ε with ε = ε + 2. Given a RUR {fI,a, fI,a,1, fI,a,X , fI,a,Y } of I, we first show how to modify the rational mapping induced by this RUR into a polynomial one.…”

“…We addressed in [4] the first phase of the above algorithm and proved that a separating linear form 1 The complexity of the isolation phase in [8,Theorem 19] is stated as e O B (d 12 +d 10 τ 2 ) but it trivially decreases to e O B (d 10 +d 9 τ ) with the recent result of Sagraloff [17] which improves the complexity of isolating the real roots of a univariate polynomial.…”

On the complexity of computing with planar algebraic curves

“…Then, in a second step, the solutions are recovered from their projections. The latter (lifting) step is relatively cheap if the projection already comes along with a parametrization of the solutions as provided by a rational univariate representation (RUR); e.g., see [29,20,12,45]. In contrast, the lifting step can be quite costly if the projection is not known to separate the solutions as given, at least in general, for a triangular decomposition.…”

On the computation of the topology of plane curves

“…Lemma 1 [7] We compute C(P ), a shear of C(P ) without vertical lines and without vertical asymptotes inÕ(d 3 τ + d 4 ) bit operations. Moreover,P is a polynomial of degree d and coefficients of bitsizeÕ(τ + d).…”