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Moving Curve Ideals of Rational Plane Parametrizations
In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations were explored in the...
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Bounds for the Condition Number of Polynomials Systems with Integer Coefficients
Polynomial systems of equations are a central object of study in computer algebra. Among the many existing algorithms for solving polynomial systems,...
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Ore Polynomials in Sage
We present a Sage implementation of Ore algebras. The main features for the most common instances include basic arithmetic and actions; GCRD and...
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Vectors and Matrices
We have already seen how vectors and matrices are represented in MATLAB in the chapter dedicated to variables, however we shall recall the notation.
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Matrix Algebra
We have already seen how vectors and matrices are represented in MATLAB in the chapter dedicated to variables, however we shall recall here the...
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Symbolic Matrix Algebra
In the preceding chapter's coverage of vector and matrix variables, we saw how to define vectors and matrices in MATLAB. At the same time, we defined...
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Polynomial root finding over local rings and application to error correcting codes
This article is devoted to algorithms for computing all the roots of a univariate polynomial with coefficients in a complete commutative Noetherian...
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Construction of Real Algebraic Numbers in Coq
This paper shows a construction in Coq of the set of real algebraic numbers, together with a formal proof that this set has a structure of discrete...
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Elliptic Curve Discrete Logarithm Problem over Small Degree Extension Fields
In 2008 and 2009, Gaudry and Diem proposed an index calculus method for the resolution of the discrete logarithm on the group of points of an...
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Multivariate Resultants in Bernstein Basis
Macaulay and Dixon resultant formulations are proposed for parametrized multivariate polynomial systems represented in Bernstein basis. It is proved...
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Complexity bounds for the rational Newton-Puiseux algorithm over finite fields
We carefully study the number of arithmetic operations required to compute rational Puiseux expansions of a bivariate polynomial F over a finite...
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Robust Algorithms For Generalized Pham Systems
We discuss the complexity of robust symbolic algorithms solving a significant class of zero–dimensional square polynomial systems with rational...
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Algebraic Cryptanalysis of the PKC’2009 Algebraic Surface Cryptosystem
In this paper, we fully break the Algebraic Surface Cryptosystem (ASC for short) proposed at PKC’2009 [3]. This system is based on an unusual problem...
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Multi-view geometry
This chapter studies the geometry of a single view, two views, three views and N views. We will distinguish an uncalibrated setting from a calibrated...
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Computing curve intersection by means of simultaneous iterations
A new algorithm is proposed for computing the intersection of two plane curves given in rational parametric form. It relies on the Ehrlich–Aberth...
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Polynomial parametrization of nonsingular complete intersection curves
In this paper we give a new method to compute a polynomial parametrization, in case it exists, of an affine nonsingular complete intersection curve....
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Implicitization of Parametric Curves via Lagrange Interpolation
A simple algorithm for finding the implicit equation of a parametric plane curve given by its parametric equations is presented. The algorithm is...
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Distance Computation from an Ellipsoid to a Linear or a Quadric Surface in IR n
Given the equations of the surfaces, our goal is to construct a univariate polynomial one of the zeroes of which coincides with the square of the...
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On the Mixed Cayley-Sylvester Resultant Matrix
For a generic n-degree polynomial system which contains n+1 polynomials in n variables, there are two classical resultant matrices, Sylvester...