Doing Algebraic Geometry with the RegularChains Library

Abstract

Abstract. Traditionally, Groebner bases and cylindrical algebraic decomposition are the fundamental tools of computational algebraic geometry. Recent progress in the theory of regular chains has exhibited efficient algorithms for doing local analysis on algebraic varieties. In this note, we present the implementation of these new ideas within the module AlgebraicGeometryTools of the RegularChains library. The functionalities of this new module include the computation of the (non-trivial) limit points of the quasi-component of a regular chain. This type of calculation has several applications like computing the Zarisky closure of a constructible set as well as computing tangent cones of space curves, thus providing an alternative to the standard approaches based on Groebner bases and standard bases, respectively. From there, we have derived an algorithm which, under genericity assumptions, computes the intersection multiplicity of a zero-dimensional variety at any of its points. This algorithm relies only on the manipulations of regular chains.

Complexity of triangular representations of algebraic sets

“…* Corresponding author an algebraic variety given that the decomposition is irredundant (Schost, 2003;Dahan and Schost, 2004), an efficient algorithm for zero-dimensional varieties (Dahan et al, 2005), and implementations (Wang, 2002;Alvandi et al, 2014). However, to the best of our knowledge, there are only a few algorithms (Gallo and Mishra, 1991;Szántó, 1999;Schost, 2003) for computing triangular decomposition with proven degree upper bounds for the output.…”

Computing strong regular characteristic pairs with Groebner bases

“…Triangular sets may be ordered according to the ranks (leading variables and degrees) and then the leading terms of their polynomials. Let the minimal triangular set contained in the reduced lexicographical (lex) Gröbner basis of a polynomial ideal (or trivially [1] if the ideal is unit) be called the W-characteristic set of the ideal. By strong regular Gröbner basis, we mean a reduced lex Gröbner basis G such that the W-characteristic set C of the ideal G is regular and sat(C) = G ; we call (C, G) a strong regular characteristic pair, or an src pair for short.…”

“…There are two families of algorithms for such regular triangular decomposition. One family of algorithms was proposed initially by Kalkbrener [17], and developed further by Moreno Maza and coauthors [24,1,8,9] with algorithmic techniques from the method of Lazard [19]. These algorithms are capable of computing regular triangular representations of the form (1).…”

“…One family of algorithms was proposed initially by Kalkbrener [17], and developed further by Moreno Maza and coauthors [24,1,8,9] with algorithmic techniques from the method of Lazard [19]. These algorithms are capable of computing regular triangular representations of the form (1). The other family of algorithms was proposed by the second author [30,29] which can compute regular zero decompositions of the form…”

“…It is easy to see that the decomposition (2) implies the representation (1). However, the representation (1) does not necessarily lead to the decomposition (2) and generators of the saturated ideals sat(T i ) in (1) are not explicitly provided. Nevertheless, for each sat(T i ) a Gröbner basis can be computed straightforwardly from T i .…”