S = QQ[x, y, z];
elementaryBasis = ideal(x+y+z, x*y+x*z+y*z, x*y*z);
saturate(elementaryBasis, x)
powerSumBasis = ideal(x+y+z, x^2+y^2+z^2, x^3+y^3+z^3);
saturate(powerSumBasis, x)
clearAll;

S = ZZ[x, y, z];
powerSumBasis = ideal(x+y+z, x^2+y^2+z^2, x^3+y^3+z^3);
saturate(powerSumBasis, x)
clearAll;

S = QQ[x, y, z, a..j, MonomialOrder => Eliminate 2];
F = a*x^3+b*x^2*y+c*x^2*z+d*x*y^2+e*x*y*z+f*x*z^2+g*y^3+h*y^2*z+
i*y*z^2+j*z^3;
partials = submatrix(jacobian matrix{{F}}, {0..2}, {0})
singularities = ideal(partials)
elimDiscr = time ideal selectInSubring(1,gens gb singularities);
elimDiscrbad=eliminate(elimDiscr,{z})
elimDiscr2 = substitute(elimDiscr, {z => 1});
elimDiscr3 = saturate(elimDiscr,z);
elimDiscr2==elimDiscr3
numgens elimDiscr2
# terms elimDiscr2_0

S = QQ[x, y, z];
I = ideal(x^5+y^3+z^3, x^3+y^5+z^3, x^3+y^3+z^5);
degree(I : saturate(I))

R = QQ[x,y,z];
M=random(R^4,R^{3:-1})
I= minors(3,M)
resol = res I
resol.dd_2
J=minors(3,resol.dd_2)
J==I
betti I
betti res I
(res ideal(x^2,y^2)).dd_2
betti res ideal(x^2,y^2)

R = QQ[t1,t2,t3,t4,s1,s2,s3,s4];
f = t1*t2*t3 + t1*t2*t4 + t1*t3*t4 + t2*t3*t4 ;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f)};
I=ideal (gradf_0-s1,gradf_1-s2,gradf_2-s3,gradf_3-s4)
J=eliminate(I,{t1,t2,t3,t4})
I2=ideal (I,f)
J2=eliminate(I2,{t1,t2,t3,t4})
I3=eliminate(ideal(I,J2),{s1,s2,s3,s4})
P=primaryDecomposition I3
W=P_1
degree W


R = QQ[t1,t2,t3,t4,s1,s2,s3,s4,
Degrees => {{1,0},{1,0},{1,0},{1,0},{0,1},{0,1},{0,1},{0,1}}];
f = t1*t2*t3 + t1*t2*t4 + t1*t3*t4 + t2*t3*t4 ;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f)};
M = matrix { {s1,s2,s3,s4}, gradf };
multidegree ideal (gradf_0-s1,gradf_1-s2,gradf_2-s3,gradf_3-s4)
multidegree minors(2,M)
multidegree ideal(gradf)
multidegree saturate( minors(2,M), ideal(gradf) )


R = QQ[t1,t2,t3,t4,s1,s2,s3,s4];
f = t1*t2*t3 + t1*t2*t4 + t1*t3*t4 + t2*t3*t4 ;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f)};
I=ideal (gradf_0-s1,gradf_1-s2,gradf_2-s3,gradf_3-s4)
J1=ideal(I,substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0}),substitute(random(1,R),{s1 => 0,s2=>0,s3=>0,s4=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0}))
degree J1
J2=ideal(I,substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0}),substitute(random(1,R),{s1 => 0,s2=>0,s3=>0,s4=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0}))
degree J2
J3=ideal(I,substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0}),substitute(random(1,R),{t1 => 0,t2=>0,t3=>0,t4=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0}))
degree J3
J4=ideal(I,substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0}),substitute(random(1,R),{t1 => 0,t2=>0,t3=>0,t4=>0}),substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0}))
degree J4
J5=saturate(J4, ideal(gradf))
degree J5
degree ideal gradf

R = QQ[t1,t2,t3,t4];
f = t1*t2*t3 + t1*t2*t4 + t1*t3*t4 + t2*t3*t4 ;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f)};
S=singularLocus ideal f
dim S
degree S


R = QQ[t1,t2,t3,t4,t5,t6,s1,s2,s3,s4,s5,s6,
Degrees => {{1,0},{1,0},{1,0},{1,0},{1,0},{1,0},{0,1},{0,1},{0,1},{0,1},{0,1},{0,1}}];
f = t1*t2*t3 + t1*t2*t4 + t1*t3*t4 + t2*t3*t4+t5*t1*t2+t5*t1*t3+t5*t1*t4+t5*t1*t6+t5*t2*t3+t5*t2*t4+t5*t2*t6+t5*t3*t4+t5*t3*t6+t5*t4*t6+t6*t1*t2+t6*t1*t3+t6*t1*t4+t6*t2*t3+t6*t2*t4+t6*t3*t4;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f),diff(t5,f),diff(t6,f)};
M = matrix { {s1,s2,s3,s4,s5,s6}, gradf };
multidegree saturate( minors(2,M), ideal(gradf) )
degree singularLocus ideal f;

R = QQ[t1,t2,t3,t4,t5,t6,s1,s2,s3,s4,s5,s6,
Degrees => {{1,0},{1,0},{1,0},{1,0},{1,0},{1,0},{0,1},{0,1},{0,1},{0,1},{0,1},{0,1}}];
f = t1*t2*t3*t4 + t1*t2*t3*t5 + t1*t2*t3*t6 + t1*t2*t4*t5+t1*t2*t4*t6+t1*t2*t5*t6+t1*t3*t4*t5+t1*t3*t4*t6+t1*t3*t5*t6+t1*t4*t5*t6+t2*t3*t4*t5+t2*t3*t4*t6+t2*t3*t5*t6+t2*t4*t5*t6+t3*t4*t5*t6;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f),diff(t5,f),diff(t6,f)};
M = matrix { {s1,s2,s3,s4,s5,s6}, gradf };
multidegree saturate( minors(2,M), ideal(gradf) )

R = ZZ[t1,t2,t3,t4,t5,t6,s1,s2,s3,s4,s5,s6]
f = t1*t2*t3*t4 + t1*t2*t3*t5 + t1*t2*t3*t6 + t1*t2*t4*t5+t1*t2*t4*t6+t1*t2*t5*t6+t1*t3*t4*t5+t1*t3*t4*t6+t1*t3*t5*t6+t1*t4*t5*t6+t2*t3*t4*t5+t2*t3*t4*t6+t2*t3*t5*t6+t2*t4*t5*t6+t3*t4*t5*t6;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f),diff(t5,f),diff(t6,f)};
I=ideal (gradf_0-s1,gradf_1-s2,gradf_2-s3,gradf_3-s4,gradf_4-s5,gradf_5-s6)
J1=ideal(I,substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1 => 0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}))
degree J1
J2=ideal(I,substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1 => 0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{t1=>0,s2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}))
degree J2

R = ZZ/14741[t1,t2,t3,t4,t5,t6,s1,s2,s3,s4,s5,s6]
f = t1*t2*t3*t4 + t1*t2*t3*t5 + t1*t2*t3*t6 + t1*t2*t4*t5+t1*t2*t4*t6+t1*t2*t5*t6+t1*t3*t4*t5+t1*t3*t4*t6+t1*t3*t5*t6+t1*t4*t5*t6+t2*t3*t4*t5+t2*t3*t4*t6+t2*t3*t5*t6+t2*t4*t5*t6+t3*t4*t5*t6;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f),diff(t5,f),diff(t6,f)};
I=ideal (gradf_0-s1,gradf_1-s2,gradf_2-s3,gradf_3-s4,gradf_4-s5,gradf_5-s6)
J1=ideal(I,substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1 => 0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}))
degree J1
J4=ideal(I,substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{s1 => 0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}))
degree J4
J5=ideal(I,substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{t1 => 0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}))
degree J5
J6=ideal(I,substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{t1 => 0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{s1=>0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}))
degree J6
J3=ideal(I,substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{s1 => 0,s2=>0,s3=>0,s4=>0,s5=>0,s6=>0}),substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}),substitute(random(1,R),{t1=>0,t2=>0,t3=>0,t4=>0,t5=>0,t6=>0}))
JJ=ideal(J3, s1-1)
degree JJ

R = ZZ/14741[t1,t2,t3,t4,t5,t6]
S = ZZ/14741[s1,s2,s3,s4,s5,s6]
f = t1*t2*t3*t4 + t1*t2*t3*t5 + t1*t2*t3*t6 + t1*t2*t4*t5+t1*t2*t4*t6+t1*t2*t5*t6+t1*t3*t4*t5+t1*t3*t4*t6+t1*t3*t5*t6+t1*t4*t5*t6+t2*t3*t4*t5+t2*t3*t4*t6+t2*t3*t5*t6+t2*t4*t5*t6+t3*t4*t5*t6;
gradf = {diff(t1,f),diff(t2,f),diff(t3,f),diff(t4,f),diff(t5,f),diff(t6,f)};
ff=map(R,S,gradf)
JJ=preimage_ff ideal(random(1,R))
degree JJ