Stable normal forms for polynomial system solving

Abstract

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

Border basis relaxation for polynomial optimization

“…a choice function that chooses as leading monomial a monomial whose coefficient is maximal among the choosable monomials as described in (Mourrain and Trébuchet, 2008).…”

“…We propose a new method which combines Lasserre's SDP relaxation approach with polynomial algebra, in order to increase the efficiency of the optimization algorithm. Border basis computations are considered for their numerical stability (Mourrain and Trébuchet, 2005;Mourrain and Trébuchet, 2008). In principle, any graded normal form technique could be used here.…”

Toric border basis

“…These structures are described effectively by a set of polynomials which represent the normal forms in the quotient structure and a method to compute the normal form of any polynomial. This family of methods includes, for instance, Gröbner basis [3,6] or border basis computation [15,18,12]. A "fixed-point" strategy is involved in these algorithms: starting with the initial set of equations, so-called S-polynomials or commutation polynomials are computed and reduced.…”

“…This monomial ordering is used to define the initial ideal associated to the ideal of the equations. The border basis approach extends Gröbner basis methods by removing the monomial ordering constraint, which may induce numerical instability when the coefficients of the polynomials are known approximately [15,18,11,13,12,16,19,10,20].…”

“…We extend the border basis approach to Laurent polynomials and show that a characterization similar to the criterion for classical border bases [15,18,19] applies in this case: namely, the commutation relations (the product of two variables should commute) and the inversion relations (the product of a variable by its inverse should be 1) can be used to check if a set of polynomials is a toric border basis in a given degree. As a new result, this characterization yields an explicit description of the first module of syzygies of a toric border basis, generalizing in a natural way results from [19]. It is an extension of Schreyer Theorem which describes generators of the first syzygy module of a Grobner basis in terms of the reduction of S-polynomials [6][Theorem 15.10].…”