# Origin of the problem: Cuspidal Robots (Wenger/Rouillier)
# [Robotic]
# Level: Middle
# Cuspidal Robots :  Resolution using Discriminant varieties
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# Equations : 3 equations in 3 variables X,z,rho depending on 3 parameters d4,d3,r2.
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# Inequalities : rho >=0 d4>0, d3>0, r2>0
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# Hypothesis : none
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# Informations from roboticians : dimension 0 for almost all the parameter's values
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# Question : number of real solutions of the following system wrt the parameter's values (cuspidal robots are caracterized by parameter's values for which there exists admissible solutions to the problem)
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[1-4*d4*d3*z^2-4*d4*d3*rho^2-8*X^3*r2*d4*z^2-8*X*r2*d4*rho^2-8*X*r2*d4*z^2+16*X*r2*d3*d4^2-2*d3^2-2*d4^2+2*r2^2+rho^4+z^4+d3^4+d4^4+r2^4+4*d4*d3^3+4*d4^3*d3+2*rho^2*z^2-2*rho^2*d3^2-2*rho^2*d4^2-2*rho^2*r2^2-2*z^2*d3^2-2*z^2*d4^2-2*z^2*r2^2+2*d3^2*r2^2+2*d4^2*r2^2-2*X^4*d3^2-2*X^4*d4^2+2*X^4*r2^2+X^4*rho^4+X^4*z^4+X^4*d3^4+X^4*d4^4+X^4*r2^4-2*X^4*rho^2+2*X^4*z^2-4*X^2*d3^2+12*X^2*d4^2+4*X^2*r2^2+2*X^2*rho^4+2*X^2*z^4+2*X^2*d3^4+2*X^2*d4^4+2*X^2*r2^4-4*X^2*rho^2+4*X^2*z^2+6*d4^2*d3^2-4*X^4*d4*d3*r2^2-16*X^3*r2*d3*d4^2+8*X^3*r2*d4*d3^2-8*X^3*r2*d4*rho^2+4*X^4*d4*d3*z^2+4*X^4*d4*d3*rho^2-2*rho^2+2*z^2-4*X^4*d4*d3^3-4*X^4*d4^3*d3+2*X^4*rho^2*z^2-2*X^4*rho^2*d3^2-2*X^4*rho^2*d4^2-2*X^4*rho^2*r2^2-2*X^4*z^2*d3^2-2*X^4*z^2*d4^2-2*X^4*z^2*r2^2+2*X^4*d3^2*r2^2+2*X^4*d4^2*r2^2+6*X^4*d4^2*d3^2+4*X^4*d4*d3+8*X^3*r2*d4+8*X^3*r2*d4^3+8*X^3*r2^3*d4+4*X^2*rho^2*z^2-4*X^2*rho^2*d3^2-4*X^2*rho^2*d4^2-4*X^2*rho^2*r2^2-4*X^2*z^2*d3^2-4*X^2*z^2*d4^2-4*X^2*z^2*r2^2+4*X^2*d3^2*r2^2+20*X^2*d4^2*r2^2-4*X^2*d4^2*d3^2+8*X*r2*d4+8*X*r2*d4^3+8*X*r2^3*d4+4*d4*d3*r2^2+8*X*r2*d4*d3^2-4*d4*d3+X^4+2*X^2,
 10*X*d4^2*r2^2-2*X*z^2*d4^2-2*X*rho^2*d3^2-2*X*rho^2*r2^2+4*X^3*d4*d3*z^2+4*X^3*d4*d3*rho^2-4*X^3*d4*d3*r2^2-12*X^2*r2*d3*d4^2+6*X^2*r2*d4*d3^2-6*X^2*r2*d4*rho^2-6*X^2*r2*d4*z^2-2*X*z^2*d3^2+2*r2^3*d4-2*X*d4^2*d3^2+6*X^2*r2*d4+2*r2*d4^3+2*X*d3^2*r2^2+X+6*X^2*r2*d4^3+6*X^2*r2^3*d4+2*r2*d4*d3^2-2*r2*d4*rho^2-2*r2*d4*z^2-4*X^3*d4*d3^3-4*X^3*d4^3*d3+2*X^3*rho^2*z^2-2*X^3*rho^2*d3^2-2*X^3*rho^2*d4^2-2*X^3*rho^2*r2^2-2*X^3*z^2*d3^2-2*X^3*z^2*d4^2-2*X^3*z^2*r2^2+2*X^3*d3^2*r2^2+2*X^3*d4^2*r2^2+6*X^3*d4^2*d3^2+4*X^3*d4*d3-2*X^3*d3^2+2*X^3*r2^2+X^3*rho^4+X^3*z^4+X^3*d3^4+X^3*d4^4+X^3*r2^4-2*X^3*rho^2+2*X^3*z^2-2*X*d3^2+6*X*d4^2+2*X*r2^2+X*rho^4+X*z^4+X*d3^4+X*d4^4+X*r2^4-2*X*rho^2+2*X*z^2-2*X^3*d4^2+4*r2*d3*d4^2-2*X*rho^2*d4^2+2*r2*d4+2*X*rho^2*z^2-2*X*z^2*r2^2+X^3,
 1-12*X*r2*d4*rho^2-12*X*r2*d4*z^2-24*X*r2*d3*d4^2+12*X^2*d4*d3*z^2+12*X^2*d4*d3*rho^2-12*X^2*d4*d3*r2^2-2*d3^2+6*d4^2+2*r2^2+rho^4+z^4+d3^4+d4^4+r2^4+2*rho^2*z^2-2*rho^2*d3^2-2*rho^2*d4^2-2*rho^2*r2^2-2*z^2*d3^2-2*z^2*d4^2-2*z^2*r2^2+2*d3^2*r2^2+10*d4^2*r2^2-6*X^2*d3^2-6*X^2*d4^2+6*X^2*r2^2+3*X^2*rho^4+3*X^2*z^4+3*X^2*d3^4+3*X^2*d4^4+3*X^2*r2^4-6*X^2*rho^2+6*X^2*z^2-2*d4^2*d3^2+12*X^2*d4*d3-2*rho^2+2*z^2+6*X^2*rho^2*z^2-6*X^2*rho^2*d3^2-6*X^2*rho^2*d4^2-6*X^2*rho^2*r2^2-6*X^2*z^2*d3^2-6*X^2*z^2*d4^2-6*X^2*z^2*r2^2+6*X^2*d3^2*r2^2+6*X^2*d4^2*r2^2+18*X^2*d4^2*d3^2+12*X*r2*d4+12*X*r2*d4^3+12*X*r2^3*d4-12*X^2*d4*d3^3-12*X^2*d4^3*d3+12*X*r2*d4*d3^2+3*X^2]