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The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms… (More)

In this paper we give an elementary approach to univariate polynomial subresultants theory. Most of the known results of subresultants are recovered, some with more precision, without using Euclidean divisions or existence of roots for univariate polynomials. The main contributions of this paper are not new results on subresultants, but rather extensions of… (More)

The calculation of threshold conditions for models of infectious diseases is of central importance for developing vaccination policies. Such models often consist of coupled systems of ordinary differential equations, and the computation of threshold conditions can be reduced to the question of the stability of the disease free equilibrium. We show how the… (More)

Symbolic methods to investigate Hopf bifurcation problems of vector fields arising in the context of algebraic biology have recently obtained renewed attention. However, the symbolic investigations have not been fully algorithmic but required a sequence of symbolic computations intervened with ad hoc insights and decisions made by a human. In this paper we… (More)

In this paper we give a semi-algebraic description of Hopf bifurcation fixed points for a given parameterized polynomial vector field. The description is carried out by use of the Hurwitz determinants, and produces a first-order formula which is transformed into a quantifier-free formula by the use of usual-quantifier elimination algorithms. We apply… (More)

In this paper we give a new projection-based algorithm for computing the topology of a real algebraic space curve given implicitly by a set of equations. Under some genericity conditions, which may be reached through a linear change of coordinates, we show that a plane projection of the given curve, together with a special polynomial in the ideal of the… (More)

We give an easy and efficient algorithm to check whether a given polynomial <i>f</i> in <i>K</i>[<i>x,y</i>] is a coordinate, where <i>K</i> be a commutative field of characteristic zero, and if so to compute a coordinate's mate of <i>f</i>. Then we treat the same problem replacing the ground field <i>K</i> by a unique factorization domain <i>A</i> of… (More)