# THIS IS THE SOLUTION OF PROBLEM 3 in Midterm 2 <<<<<<<<<<<<<<<<<<<<<restart;with(Groebner);F:=[x^2+2*y^2-y-2*z,x^2-8*y^2+10*z-1,x^2-7*y*z];

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G:=gbasis(F,plex(x,y,z));

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_EnvExplicit:=true; ycoord:=solve(G[2],{y}); xcoord:=solve(G[3],{x});

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rootlist:=[evalf(solve(G[1]))];

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for z in rootlist do print('z-coordinate',z, 'y-coordinate',ycoord, 'x-coordinate', xcoord);od:

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# Below is the SOLUTION FOR PROBLEM 2B in MIDTERM 2 <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<restart; with(Groebner); F:=[x^3+x+1, y^3+y+1, z^3+z+1, w^3+w+1, (x-y)*( y-z)*( x-z)*(x-w)*(y-w)*(z-w)];

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# We see below the number of solutions is eighty one because solutions for x,y,z,w are independent of each othergbasis(F,tdeg(x,y,z,w));

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# Now we do the solution of PROBLEM 2A in MIDTERM 2 <<<<<<<<<<<<<<<<<<<<restart;with(Groebner);

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F:=[x^2+y^2-1,x^2+y^2+(z-1)^2-4];

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p:=x^2+(1/2)*y^2*z-z-1;

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normalf(p,gbasis(F,plex(x,y,z)),plex(x,y,z));

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