Agnes Szanto

Tulay Ayyildiz Akoglu, Jonathan D. Hauenstein, Agnes Szanto

This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is not a continuous property. Our certification is based on hybrid symbolic-numeric methods to compute the exact "rational univariate representation" (RUR) of a component of the input system from approximate roots. For overdetermined polynomial systems with simple roots, we compute an initial RUR from approximate roots. The accuracy of the RUR is increased via Newton iterations until the exact RUR is found, which we certify using exact arithmetic. Since the RUR is well-constrained, we can use it to certify the given approximate roots using alpha-theory. To certify isolated singular roots, we use a determinantal form of the "isosingular deflation", which adds new polynomials to the original system without introducing new variables. The resulting polynomial system is overdetermined, but the roots are now simple, thereby reducing the problem to the overdetermined case. We prove that our algorithms have complexity that are polynomial in the input plus the output size upon successful convergence, and we use worst case upper bounds for termination when our iteration does not converge to an exact RUR. Examples are included to demonstrate the approach.

Itnuit Janovitz-Freireich, Lajos Ronyai, Agnes Szanto

We present a method based on Dickson's lemma to compute the "approximate radical" of a zero dimensional ideal I in C[x1, . . ., xm] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is "global" in the sense that it does not require any local approximation of the zero clusters: it reduces the problem to the computation of the numerical nullspace of the so called "matrix of traces", a matrix computable from the generating polynomials of I. To compute the numerical nullspace of the matrix of traces we propose to use Gaussian elimination with pivoting or singular value decomposition. We prove that if I has k distinct zero clusters each of radius at most epsilon in the max-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically bounded by epsilon^2. We also show that the (k+1)-th singular value of the matrix of traces is O(epsilon^2). The resulting approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically bounded by epsilon^2. In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood.