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 [MT'05]. This general border basis
algorithm weakens the monomial ordering requirement for \grob bases
computations. It is up to date the most general setting for representing
quotient algebras, embedding into a single formalism Gr\"obner 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
In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finte. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of [17] are efficient and numerically stable for computing complex roots, algorithms based on moment matrices [12] allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Gröbner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.J.B. Lasserre, LAAS,
International audienceThis report describes a new method for computing the normal form of a polynomial modulo a zero-dimensional ideal $I$. We give a detailed description of the algorithm, a proof of its correctness, and finally experimentations on classical benchmark polynomial systems. The method that we propose can be thought as an extension of both the Gröbner basis method and the Macaulay construction. As such it establishes a natural link between these two methods. We have weaken the monomial ordering requirement for Gröbner bases computations, which allows us to construct new type of representations for the associated quotient algebra. This approach yields more freedom in the linear algebra steps involved, which allows us to take into account numerical criteria while performing the symbolic steps. This is a new feature for a symbolic algorithm, which has an important impact on the practical efficiency, as it is illustrated by the experiments at the end of the paper
In this paper, a novel analytical blind single-input single-output (SISO) identification algorithm is presented, based on the noncircular second-order statistics of the output. It is shown that statistics of order higher than two are not mandatory to restore identifiability. Our approach is valid, for instance, when the channel is excited by phase shift keying (PSK) inputs. It is shown that the channel taps need to satisfy a polynomial system of degree 2 and that identification amounts to solving the system. We describe the algorithm that is able to solve this particular system entirely analytically, thus avoiding local minima. Computer results eventually show the robustness with respect to noise and to channel length overdetermination. Identifiability issues are also addressed.
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