On the complexity of solving bivariate systems

Abstract

We give an algorithm for solving bivariate polynomial systems over either k(T )[X, Y ] or Q[X, Y ] using a combination of lifting and modular composition techniques.

Uniform Determinantal Representations

“…Such an approach is thus not efficient for polynomials of large degree. We remark that there also exist probabilistic symbolic algorithms (see, e.g., [23,26]) that aim for a smaller complexity, such as one not much higher than O(d 4 ).…”

Fast arithmetic for the algebraic closure of finite fields

“…We use baby steps / giant steps techniques from [28] (inspired by Brent and Kung's algorithm) to compute a, reducing the problem to polynomial matrix multiplication. Let n = m+n−1, p = √ n and q = n /p , so that n ≤ n ≤ 2n − 1 and p q √ n. For baby steps, we compute the polynomials Ti = T i mod R, which have degree at most mn−1; we write Ti = 0≤j<n T i,j z jm , with T i,j of degree less than m, and build the polynomial matrix M T with entries T i,j .…”

Newton iteration for lexicographic Gröbner bases in two variables

“…Solving bivariate polynomial equations plays an important role in algorithms for computational topology or computer graphics. As a result, there exists a large body of work dedicated to this question, using symbolic, numeric or mixed symbolic-numeric techniques [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16].…”

“…This paper focuses on those cases where the ideal I is not radical (that is, where some points p ∈ V (I) are singular), with the intent of computing the local structure at such points. If the sole interest is to find V (I), then our approach is unnecessarily complex: the algorithms in [9,12] use Newton iteration to compute a set-theoretic description of the solutions in an efficient manner.…”