A Fast Algorithm for Computing Macaulay Null Spaces of Bivariate Polynomial Systems

.As a crucial first step towards finding the (approximate) common roots of a (possibly overdetermined) bivariate polynomial system of equations, the problem of determining an explicit numerical basis for the right null space of the system's Macaulay matrix is considered. If \(d_{\Sigma }\in \mathbb{N}\) denotes the total degree of the bivariate polynomials of the system, the cost of computing a null space basis containing all system roots is \(\mathcal{O}( d^6_{\Sigma })\) floating point operations through standard numerical algebra techniques (e.g., a singular value decomposition, rank-revealing QR-decomposition). We show that it is actually possible to design an algorithm that reduces the complexity to \(\mathcal{O}(d^5_{\Sigma })\). The proposed algorithm exploits the Toeplitz structures of the Macaulay matrix under a nongraded lexicographic ordering of its entries and uses the low displacement rank properties to efficiently convert it into a Cauchy-like matrix with the help of fast Fourier transforms. By modifying the classical Schur algorithm with total pivoting for Cauchy-like matrices, a compact representation of the right null space is eventually obtained from a rank-revealing LU-factorization. Details of the proposed method, including numerical experiments, are fully provided for the case wherein the polynomials are expressed in the monomial basis. Furthermore, it is shown that an analogous fast algorithm can also be formulated for polynomial systems expressed in the Chebyshev basis.KeywordsMacaulay matricespolynomials systemsrank-revealing LU-factorizationslow displacement rank matricesSchur algorithmMSC codes15A6915A23