An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

Computing Monodromy via Continuation Methods on Random Riemann Surfaces

International audienceWe consider a Riemann surface $X$ defined by a polynomial $f(x,y)$ of degree $d$, whose coefficients are chosen randomly. Hence, we can suppose that $X$ is smooth, that the discriminant $\delta(x)$ of $f$ has $d(d-1)$ simple roots, $\Delta$, and that $\delta(0) \neq 0$ i.e. the corresponding fiber has $d$ distinct points $\{y_1, \ldots, y_d\}$. When we lift a loop 0 \in \gamma \subset \Ci - \Delta by a continuation method, we get $d$ paths in $X$ connecting $\{y_1, \ldots, y_d\}$, hence defining a permutation of that set. This is called monodromy. Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of $\Delta$. Multiplying families of ''neighbor'' transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups. Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.On considere une surface de Riemann dont l'equation f(x,y)=0 est un polynome dont les coefficients sont des variables aleatoires Gaussiennes standards, ainsi que sa projection p sur l'axe des x. Puis on etudie et calcule des generateurs du groupe de monodromie correspondant a p

Exact Symbolic-Numeric Computation of Planar Algebraic Curves

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry. We have implemented our algorithms as prototypical contributions to the C++-project CGAL. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, LGP and Maple's Isolate for polynomial system solving, and CGAL's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011. arXiv admin note: substantial text overlap with arXiv:1010.1386 and arXiv:1103.469

Factoring bivariate polynomials using adjoints

One relates factorization of bivariate polynomials to singularities of projective plane curves. One proves that adjoint polynomials permit to solve the recombinations of the modular factors induced by the absolute and rational factorizations, and so without using Hensel's lifting. One establishes in such a way the relations between the algorithm of Duval-Ragot (locally constant functions) and of Ch\`eze-Lecerf (lifting and recombinations), and one shows that a fast computation of adjoint polynomials leads to a fast factorization. The proof is based on cohomological sequences and residue theory.Comment: 22 pages, 2 figures. Extended version of arXiv.1201.578

Computing Puiseux series : a fast divide and conquer algorithm

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the singular parts of all Puiseux series of $F$ above $X = 0$ in less than $\tilde{\mathcal{O}}(D\delta)$ operations in $\mathbb{K}$, where $\delta$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of $F$ in $\mathbb{K}[[X]][Y ]$ up to an arbitrary precision $X^N$ with $\tilde{\mathcal{O}}(D(\delta + N ))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $\tilde{\mathcal{O}}(D^3)$ arithmetic operations and, if $\mathbb{K} = \mathbb{Q}$, with $\tilde{\mathcal{O}}((h+1)D^3)$ bit operations using a probabilistic algorithm, where $h$ is the logarithmic heigth of $F$.Comment: 27 pages, 2 figure

On the computation of the topology of plane curves

International audienceLet P be a square free bivariate polynomial of degree at most d and with integer coefficients of bit size at most t. We give a deterministic algorithm for the computation of the topology of the real algebraic curve definit by P, i.e. a straight-line planar graph isotopic to the curve. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular and critical points of the projection wrt the X axis in $\tilde{O} (d^6 t)$ bit operations where $\tilde{O}$ means that we ignore logarithmic factors in $d$ and $t$. Combined to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of the curve in $\tilde{O} (d^6 t + d^7)$ bit operations