Solving some overdetermined polynomial systems
M Giusti, Éric Schost · 1999
We propose a strategy to obtain approsin~ate solutions of an ovcrdctcrminecl consistent polynomial system of which we are only given an approximation.These systems will bc supposc:d to be instantiations of some polynomials Fl: . . .F,,.+I E k[X, :c] on the X variables: where k C C is an effective field: .c= (xl, : x,,) arc t.hc unknowns a.ntl X is to be seen as a set, of parameters.For an arbitrary choice of A; this system is generally inconsistent.M'c first propose hypothcscs under which the set of X where the systcni is consistent is an hypcrsurfacc of the space of paramclt.clrs.In the second part, wc use t.he algorithm for gcomctric resolution given in [7] in our particular setting, to give a theoretical polynomial-time resolution algorithm.Finally, we apply our st.rategv to the csaruple of an overconst,rainc:d parallel manipulator; where an est.ra m(xsurc is adjoined.-4 resolution is comput,ed in Magma t,hat demonstrates the feasibility of the method. .,