--- Matrix completion
restart
R=QQ[x12,x13,x14,x15,x23,x24,x25,x34,x35,x45,x11,x22,x33,x44,x55,MonomialOrder=>Lex]
M=matrix{{x11,x12,x13,x14,x15},{x12,x22,x23,x24,x25},{x13,x23,x33,x34,x35},{x14,x24,x34,x44,x45},{x15,x25,x35,x45,x55}}
J=minors(3,M);
isPrime J
degree J
codim J
transpose gens J
transpose gens gb J -- one equation indep of diagonal elements.
eliminate({x11,x22,x33,x44,x55},J) -- gives directly the equation



-- Lagrange multiplier
restart
R=QQ[l,x,y,z,MonomialOrder=>GRevLex] -- l>x>y>z
f=x^3+2*x*y*z-z^2 -- function
h=x^2+y^2+z^2-1 -- constraint
J=ideal jacobian matrix{{f+l*h}}
Js=gens gb J 
transpose Js

-- Héron formula
restart
R=QQ[x,y,s,a,b,c,MonomialOrder=>Lex]
I= ideal (b^2-(a-x)^2-y^2,c^2-x^2-y^2,2*s-a*y)
M=transpose gens gb I
E=M_(0,0)
factor (E-16*s^2)


-- 3-color maps

R=QQ[x1,x2,x3,x4,MonomialOrder=>Lex]
f1=x1^3-1
f2=x2^3-1
f3=x3^3-1
f4=x4^3-1
Q12=(factor(x1^3-x2^3))#1#0
Q13=(factor(x1^3-x3^3))#1#0
Q14=(factor(x1^3-x4^3))#1#0
Q23=(factor(x2^3-x3^3))#1#0
Q24=(factor(x2^3-x4^3))#1#0
Q34=(factor(x3^3-x4^3))#1#0
I=ideal(f1,f2,f3,f4,Q12,Q13,Q14,Q23,Q24,Q34)
transpose gens gb I


R=QQ[x1,x2,x3,MonomialOrder=>Lex]
f1=x1^3-1
f2=x2^3-1
f3=x3^3-1
Q12=(factor(x1^3-x2^3))#1#0
Q13=(factor(x1^3-x3^3))#1#0
Q23=(factor(x2^3-x3^3))#1#0
I=ideal(f1,f2,f3,Q12,Q13,Q23)
transpose gens gb I

-- your own more complex map
restart
R=QQ[x1,x2,x3,x4,x5,MonomialOrder=>Lex]
f1=x1^3-1
f2=x2^3-1
f3=x3^3-1
f4=x3^3-1
f5=x3^3-1
Q12=(factor(x1^3-x2^3))#1#0
Q13=(factor(x1^3-x3^3))#1#0
Q14=(factor(x1^3-x4^3))#1#0
Q23=(factor(x2^3-x3^3))#1#0
Q25=(factor(x2^3-x5^3))#1#0
Q34=(factor(x3^3-x4^3))#1#0
Q35=(factor(x3^3-x5^3))#1#0
Q45=(factor(x4^3-x5^3))#1#0
I=ideal(f1,f2,f3,f4,f5,Q12,Q13,Q14,Q23,Q25,Q34,Q35,Q45)
transpose gens gb I