In this paper, we present a new method for solving square polynomial systems with no zero at infinity. We analyze its complexity, which indicates substantial improvements, compared with the previously known methods for solving such systems. We describe a framework for symbolic and numeric computations, developed in C++, in which we have implemented this algorithm. We mention the techniques that are involved in order to build efficient codes and compare with existing softwares. We end by some applications of this method, considering in particular an autocalibration problem in Computer Vision and an identification problem in Signal Processing, and report on the results of our first implementation.

This paper presents characterizations of border bases of zero-dimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of the following equivalent properties: special generation, generation of the border form ideal, confluence of the corresponding rewrite relation, reduction of S-polynomials to zero, and lifting of syzygies. The last characterization relies on a detailed study of the relative position of the border terms and their syzygy module. In particular, a border prebasis is a border basis if and only if all fundamental syzygies of neighboring border terms lift; these liftings are easy to compute. Key words: border basis, division algorithm, syzygy module 1

Consider a general polynomial system S in x1,..., xn of degree q and its corresponding vector of monomials of degree less than or equal to q. The system can be written as M0 · [x q 1, x q−1

real radical ideals

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We are interested in computing e#ectively cylinders through 5 points, and in other problems involved in metrology. In particular, we consider the cylinders through 4 points with a #x radius and with extremal radius. For these di#erent problems, we give bounds on the number of solutions and exemples show that these bounds are optimal. Finally,we describe two algebraic methods which can be used here to solve e#ciently these problems and some experimentation results.

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal I. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [29]. This general border basis algorithm weakens the monomial ordering requirement for Gröbner bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Gröbner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper.

This paper presents several algorithms that compute border bases of a zero-dimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based on a generic algorithm by Bernard Mourrain originally designed for computing an ideal basis that need not be a border basis. Our fully detailed algorithm computes a border basis of a zerodimensional ideal from a given set of generators. To obtain concrete instructions we appeal to a degree-compatible term ordering σ and hence compute a border basis that contains the reduced σ-Gröbner basis. We show an example in which this computation actually has advantages over Buchberger’s algorithm. Moreover, we formulate and prove two optimizations of the Border Basis Algorithm which reduce the dimensions of the linear algebra subproblems.

The Leibniz formula, for the divided difference of a product, and Opitz's formula, for the divided difference table of a function as the result of evaluating that function at a certain matrix, are shown to be special cases of a formula available for the coefficients, with respect to any basis, of an `ideal' or

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