Irreducible Decomposition of Curves

Abstract

In this paper, we propose a fast semi-numerical algorithm for computing all irreducible branches of a curve in C τ defined by polynomials with rational coefficients, we also treat the case of a non-reduced curve. Our algorithm does not requires a prior projection procedure which, in many cases, is difficult to achieve. It relies on a fine analysis of a generic "fat" section of the curve. Our approach could be applied to more general situations, it generalizes our previous study on absolute factorization of polynomials.

Absolute polynomial factorization in two variables and the knapsack problem

“…This was proposed by A. Galligo and D. Rupprecht in [16], [8]. Then absolute factorization is reduced to finding the minimal zero sum relations between a set of approximated numbers bi, i = 1 to n such that P n i=1 bi = 0, (see also [17]).…”

“…Another kind of algorithms use topological properties of C 2 , Newton approximation or so called homotopy methods and for which floating point approximations are better suited, e.g. algorithms of Sasaki and coworkers (see [15], [17], [18]), Galligo and coworkers (see [3], [7], [8]), Sommese-Verschelde-Wampler (see [19], [20]). Here we follow a symbolic-numeric method to get an absolute factorization.…”

Factoring Solution Sets of Polynomial Systems in Parallel

“…Finally, we observed that the monodromy breakup algorithm makes the most gains in the £rst loops, often halving the number of irreducible factors in one step, but then spends several loops to £nd the connections for factors of small degree. A hybrid strategy, combining alternative methods [7,15] based on the linear trace will further increase the performance of the factorization algorithm.…”

“…For example, x 2 + y 2 − 1 is irreducible, whereas x 2 − y 2 can be written as the product of x − y and x + y. The need for a polynomial time algorithm for the approximate multivariate polynomial factorization expressed in [12] received a lot of research attention [4,5,7,8,11,15,20].…”

“…The use of the linear trace is a key ingredient in many factorization algorithms, see for example [4,7,15,18,20]. While for low degree solution sets, a good approach (as done in [7]) is to enumerate all possible partitions and then to apply the linear trace test, for high degrees this approach becomes prohibitively expensive. We can predict a breakup by generating loops around singularities, exploiting the monodromy group action.…”