ta[x_] := Together[Apart[x]];
te[x_] := Together[Expand[x]];
fe[x_] := Factor[Expand[x]];
ce[x_] := ComplexExpand[x];
mp[A_,p_]:=MatrixPower[A,p];
mf[A_] := MatrixForm[A];
simp[x_]:= Simplify[x];
fs[x_]:= FullSimplify[x];
rat[x_]:=Rationalize[x];

eq0[x_] := Module[{y}, f[y_] := y == 0; Map[f, x]]

minor[M_,i_,j_] :=
Table[M[[Drop[Table[k,{k,Length[M]}],{i}][[m]],
     Drop[Table[k,{k,Length[M]}],{j}][[n]]]]
,{m,Length[M]-1},{n,Length[M]-1}];

cofactor[M_] :=
Transpose[Table[(-1)^(i+j)*Det[minor[M,i,j]],{i,Length[M]},{j,Length[M]}]];

repose[M_] :=
Array[m,Dimensions[M]]/.
  Flatten[Table[m[i,j]->M[[Dimensions[M][[2]]-j+1,Dimensions[M][[1]]-i+1]]
,{i,Dimensions[M][[1]]},{j,Dimensions[M][[2]]}],1];

funcEval[F_,x_,y_] :=
Module[{z},z=x; F/.Table[z[[i]]->y[[i]],{i,Length[x]}]];

norm[x_,n_] := If[x[[n]]!=0,x/x[[n]],x,x/x[[n]]];
homPolyDeg[f_, x_] := Exponent[
  Expand[f][[1]] /. Table[x[[i]] -> y, {i, Length[x]}]
  , y];

jacob[f_,x_] := Module[{y},y=x;
Table[D[f[[i]],y[[j]]],{i,Length[f]},{j,Length[x]}]];
jacobEval[f_,x_,y_] := funcEval[jacob[f,x],x,y];

diff[F_,x_] := Module[{y},y=x;Table[D[F, y[[i]]],{i,Length[x]}]];
diffEval[F_,x_,y_] := funcEval[diff[F,x],x,y];

cross[f_, x_] := diff[
                   Det[Append[
                     jacob[f, x]
                   , Table[U[i], {i, Length[x]}]]]
                 , Table[U[i], {i, Length[x]}]];

reId[n_] := Table[IdentityMatrix[n][[n-i]],{i,0,n-1}]
reDiff[F_,x_] :=reId[Length[x]].diff[F,x]

wedge1form1form[f_,g_] := Cross3[A,B]/.{A->f,B->g};
wedge1form1formEval[f_,g_,x_,y_] := funcEval[Wedge1form1form[f,g],x,y];

numfunc[f_,x_,n_] := N[f[N[x,n]],n];

hPlaneOnDimpoints[a_,x_] := Collect[ta[ Det[Insert[a,x,1]] ] ,x];

pointOnDimHPlanes[l_,x_,normVar_] :=
ta[norm[x/.Solve[Table[l[[i]]==0,{i,Length[l]}],x][[1]],normVar]]//Quiet;

eucNorm[x_] := Sum[x[[i]]^2,{i,Length[x]}]/Sqrt[Sum[x[[i]]^2,{i,Length[x]}]];

hermprod[x_,y_] := ta[ComplexExpand[Conjugate[x].y]];

hermform[P_] := ta[P.ComplexExpand[Conjugate[Transpose[P]]]];

linNorm[l_,x_,n_] :=
Module[{y},y=x;
  ta[(Collect[Apart[l],y]/Coefficient[Collect[Apart[l],y],y[[n]]])]
];

linParam[l_,n_,x_] := x/.Solve[l==0,x[[n]]][[1]];

ratDenom[a_, x_, y_] := a /.
    (x + Sqrt[y]) -> (x^2 - y)/(x - Sqrt[y])

crossRatio[a_,b_,c_,d_]:=((a-c)*(b-d))/((a-d)*(b-c))
mob3Pts[{a_,b_,c_}, {A_,B_,C_}]:=
  fs[w/.Solve[crossRatio[z,a,b,c]==crossRatio[w,A,B,C],w][[1]]]
C2rect[z_] := {z//Re//ce,z//Im//ce}
rect2C[x_] := x[[1]]+I*x[[2]] 
s2C[x_] := x[[1]]/(1-x[[3]]) + I*x[[2]]/(1-x[[3]])
C2s[z_] :={2 Re[z],2 Im[z],(Re[z]^2 + Im[z]^2 - 1)}/(Re[z]^2 + Im[z]^2 + 1)

circCent3ptC[p_,q_,r_] := (Abs[p]^2 (r - q) + Abs[q]^2 (p - r) + Abs[r]^2 (q - p))/
                         (Conjugate[p] (r - q) + Conjugate[q] (p - r) + Conjugate[r] (q - p)) // ce;

circParam3ptC[p_, q_, r_][t_] := (p - circCent3ptC[p, q, r]) E^(I*t) + circCent3ptC[p, q, r]//ce;

circ3ptPlot[p_, q_, r_][T_] := ParametricPlot[C2rect[circParam3ptC[p, q, r][t]],{t, 0, T}]

\!\(permAct[n_, v_] := 
    Table[x\_i, {i, n}] /. \[IndentingNewLine]Flatten[
        Table[\[IndentingNewLine]If[
            i < Length[
                v[\([j]\)]], \[IndentingNewLine]x\_\(v[\([j, i + 1]\)]\) -> 
              x\_\(v[\([j, i]\)]\), \[IndentingNewLine]x\_\(v[\([j, 1]\)]\) \
-> x\_\(v[\([j, Length[v[\([j]\)]]]\)]\)\[IndentingNewLine]]\
\[IndentingNewLine], {j, Length[v]}, {i, Length[v[\([j]\)]]}]]\)

permMap[n_,v_,x_] := Table[x[[i]],{i, n}] /.
    Flatten[Table[
        If[i < Length[v[[j]]],
          x[[v[[j, i + 1]]]] -> x[[v[[j, i]]]],
          x[[v[[j, 1]]]] -> x[[v[[j, Length[v[[j]]]]]]]
        ]
    ,{j, Length[v]}, {i, Length[v[[j]]]}]]

(* Invariant theoretic *)

genExpress[f_,gen_,invar_,elimvar_] := 
GroebnerBasis[Union[{f},Table[invar[[i]] - gen[[i]], {i, Length[invar]}]],elimvar][[1]]

hess[F_, x_] := Table[D[F, x[[i]], x[[j]]], {i, Length[x]}, {j, Length[x]}];

bordHess[F_, G_, x_] := Append[Transpose[
  Append[
    Transpose[hess[F, x]]
   , Table[D[G, x[[i]]], {i, Length[x]}]
]], Append[Table[D[G, x[[i]]], {i, Length[x]}], 0]];

cross3[{a1_,a2_,a3_},{b1_,b2_,b3_}]:=
    {a2 b3 - a3 b2, a3 b1 - a1 b3, a1 b2 - a2 b1};

sidehill3[f_,g_]:= Together[Cross3[D3[f],D3[g]]];

critical3[f_] := Jacobian3[f[[1]],f[[2]],f[[3]]];

equalityTest[x_,y_]:= ta[x]===ta[y];

invarianceTest[f_,g_]:= equalityTest[f,Transform3[f,g]];
equivarianceTest[f_,g_]:= equalityTest[Transform3[f,g],Transform3[g,f]];