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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.

MSC:
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
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