Solving some overdetermined polynomial systems

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. . ,