Real solutions to systems of polynomial equations and parameter continuation. (English) Zbl 1309.65056

Summary: Given a parameterized family of polynomial equations, a fundamental question is to determine upper and lower bounds on the number of real solutions a member of this family can have and, if possible, compute where the bounds are sharp. A computational approach to this problem was developed by P. Dietmaier [in: Advances in robot kinematics: analysis and control. 6th symposium, Strobl/Salzburg, Austria, June/July 1998. Dordrecht: Kluwer Academic Publishers. 7–16 (1998; Zbl 0953.70006)] who used a local linearization procedure to move in the parameter space to change the number of real solutions. He used this approach to show that there exists a Stewart-Gough platform that attains the maximum of forty real assembly modes. Due to the necessary ill-conditioning near the discriminant locus, we propose replacing the local linearization near the discriminant locus with a homotopy-based method derived from the method of gradient descent arising in optimization. This new hybrid approach is then used to develop a new result in real enumerative geometry.


65H10 Numerical computation of solutions to systems of equations
68W30 Symbolic computation and algebraic computation
14Q99 Computational aspects in algebraic geometry
14P99 Real algebraic and real-analytic geometry


Zbl 0953.70006
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