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 known. This is due to the avoiding, through a sufficiently good change of coordinates, of real root computations on polynomials with coefficients in a simple real algebraic extension of ‫ޑ‬ to deal with the critical points of the considered curve. In fact, one of the main features of this algorithm is that its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve.

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 techniques from the theory of sub-resultant sequences and of Gröbner bases to come up with efficient reductions, which lead to quantifier elimination questions that can often be handled by existing quantifier elimination packages.We could implement the algorithms for the conditions on Hopf bifurcations by combining the computer algebra system Maple with packages for quantifier elimination using a Java-based component architecture recently developed by the second author. In addition to some textbook examples we applied our software system to an example discussed in a recent research paper.

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 discuss the use of algebraic and logical methods to reduce questions on the existence of Hopf bifurcations in parameterized polynomial vector fields to quantifier elimination problems over the reals combined with the use of the quantifier elimination over the reals and simplification techniques available in REDLOG. We can reconstruct most of the results given in the literature within a few seconds of computation time and extend the investigations on these systems to previously not analyzed related systems. Especially we discuss cases in which one suspects that no Hopf bifurcation fixed point exists for biologically relevant values of parameters and system variables. Here we focus on logical and algebraic techniques of finding subconditions being inconsistent with the hypothesis of the existence of Hopf bifurcation fixed points.

The calculation of threshold conditions for models of infectious diseases is of central importance for developing vaccination policies. These models are often coupled systems of ordinary differential equations, in which case the computation of threshold conditions can be reduced to the question of stability of the disease-free equilibrium. This paper shows how computing threshold conditions for such models can be done fully algorithmically using quantifier elimination for real closed fields and related simplification methods for quantifier-free formulas. Using efficient quantifier elimination techniques for special cases that have been developed by Weispfenning and others, we can can also compute whether there are ranges of parameters for which sub-threshold endemic equilibria exist.

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Abstract. A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any K-derivation of K [x, y], where K is a commutative field of characteristic zero, is a polynomial ring in one variable over K. In this paper we give an elementary proof of this theorem and show that it remains true if we replace K by any unique factorization domain of characteristic zero.

In this paper we describe an algorithm for computing the dual of a projective plane curve. The algorithm requires no extension of the field of coefficients of the curve and runs in polynomial time.