In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f = (f1, ..., fN) ∈ C[x1, ..., xn]N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α-theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist - sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, - powerful randomized algorithms for computing triangular decompositions using Hensel lifting in the zero-dimensional case and for irreducible varieties. However, in the general case, most of the algorithms computing triangular decompositions produce embedded components, which makes it impossible to directly apply the intrinsic degree bounds. This, in turn, is an obstacle for efficiently applying Hensel lifting due to the higher degrees of the output polynomials and the lower probability of success. In this paper, we give an algorithm to compute an irredundant triangular decomposition of an arbitrary algebraic set W defined by a set of polynomials in C[x1, x2, ..., xn]. Using this irredundant triangular decomposition, we are able to give intrinsic degree bounds for the polynomials appearing in the triangular sets and apply Hensel lifting techniques. Our decomposition algorithm is randomized, and we analyze the probability of success.

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This con- struction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables. We show that the roots of this new system include the original singular root but now with multiplicity one, and the new variables uniquely determine the multiplicity structure. Both constructions are 'exact' in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system.

In this tutorial we will focus on the solution of polynomial systems given with inexact coefficients using hybrid symbolic-numeric methods. In particular, we will concentrate on systems that are over-constrained or have roots with multiplicities. These systems are considered ill-posed or ill-conditioned by traditional numerical methods and they try to avoid them. On the other hand, traditional symbolic methods are not designed to handle inexactness. Ill-conditioned polynomial equation systems arise very frequently in many important applications areas such as geometric modeling, computer vision, fluid dynamics, etc.

Let f1,..., fs be a system of polynomials in K[x1,..., xm] generating a zero-dimensional ideal I , where K is an arbitrary algebraically closed field. Assume that the factor algebra A = K[x1 , . . . , xm]/I is Gorenstein and that we have a bound delta > 0 such that a basis for A can be computed from multiples of f1,..., fs of degrees at most delta. We propose a method using Sylvester or Macaulay type resultant matrices of f1,..., fs and J , where J is a polynomial of degree delta generalizing the Jacobian, to compute moment matrices, and in particular matrices of traces for A. These matrices of traces in turn allow us to compute a system of multiplication matrices {Mxi|i = 1,..., m} of the radical of I, following the approach in the previous work by Janovitz-Freireich, Ronyai and Szanto. Additionally, we give bounds for delta for the case when I has finitely many projective roots.

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 Gauss elimination with pivoting, and we prove that if I has k distinct zero clusters each of radius at most ε in the ∞-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to ε2. We also prove that the computed approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically equal to ε2. 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.

In this paper we give an elimination algorithm for differential difference polynomial systems. We use the framework of a generalization of Ore algebras, where the independent variables are non-commutative. We prove that for certain term orderings, Buchberger's algorithm applied to differential difference systems terminates and produces a Gröbner basis. Therefore, differential-difference algebras provide a new instance of non-commutative graded rings which are effective Gröbner structures.