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**1 - 5**of**5**### Doing Algebraic Geometry with the RegularChains Library

, 2014

"... 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 implem ..."

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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 function-alities of this new module include the computation of the (non-trivial) limit points of the quasi-component of a regular chain. This type of cal-ulation 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.

### Regular Chains under Linear Changes of Coordinates and Applications

"... Abstract. Given a regular chain, we are interested in questions like computing the limit points of its quasi-component, or equivalently, com-puting the variety of its saturated ideal. We propose techniques relying on linear changes of coordinates and we consider strategies where these changes can be ..."

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Abstract. Given a regular chain, we are interested in questions like computing the limit points of its quasi-component, or equivalently, com-puting the variety of its saturated ideal. We propose techniques relying on linear changes of coordinates and we consider strategies where these changes can be either generic or guided by the input. 1

### A Standard Basis Free Algorithm for Computing the Tangent Cones of a Space Curve

"... We outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a standa ..."

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We outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a standard basis.