with(Groebner);
with(ArrayTools);
with(PolynomialIdeals);
with(ListTools);

Extendd := proc(A, B) 
RETURN(Array([op(convert(A, list)), op(convert(B, list))])); 
end proc;

Appendd := proc(A, h) 
RETURN(Array([op(convert(A, list)), h])); 
end proc;

Removee := proc(A, h) 
local B; 
B := convert(A, list); 
B := subsop(h = NULL, B); 
RETURN(convert(B, Array)); 
end proc;

Lt := proc(f, t) 
RETURN(LeadingTerm(f, t)[1]*LeadingTerm(f, t)[2]); 
end proc;

Division := proc(f, L, t) 
local p, flag, i, r; global count; 
p := f; r := 0; 
while p <> 0 do 
i := 1; flag := false; 
while i <= ArrayNumElems(L) and flag = false do 
if divide(Lt(p, t), Lt(L[i], t), 'q') then 
p := expand(p - q*L[i]); 
count := count + 1; 
flag := true; 
else 
i := i + 1; 
end if; 
end do; 
if flag = false then 
r := r + Lt(p, t); 
p := p - Lt(p, t); 
end if; 
end do; 
RETURN(r); 
end proc;

Sort := proc(L, t) 
RETURN(sort(L, (b, a) -> TestOrder(a[3], b[3], t))); 
end proc;

Pair := proc(G, LtG, g, t) 
local P, A, A1, a, i, j, u, n, C; 
global pount, pk; 
P := Array([]); 
u := LeadingMonomial(g, t); 
n := ArrayNumElems(LtG); 
A := Array([seq(lcm(LtG[j], u)/u, j = 1 .. n)]); 
A1 := Array([op({op(Basis(convert(A, list), t))} minus {seq(LtG[j], j = 1 .. n)})]); 
pk := pk + n - ArrayNumElems(A1); 
for a in A1 do 
member(a, A, 'q'); 
P := Appendd(P, Array([g, G[q], lcm(u, LtG[q])])); 
pount := pount + 1; 
end do; 
RETURN(P); 
end proc;

PairS := proc(g, t) 
local P, A, A1, a, i, j, u, na, C; 
global pount, pk, no, nb, G, LtG, str, n, GO, GW, m; 
P := Array([]); 
u := LeadingMonomial(g, t); 
na := ArrayNumElems(LtG); 
A := Array([seq(lcm(LtG[j], u)/u, j = 1 .. na)]); 
A1 := Array([op({op(Basis(convert(A, list), t))} minus {seq(LtG[j], j = 1 .. na)})]); 
pk := pk + na - ArrayNumElems(A1); 
for a in A1 do member(a, A, 'q'); 
if 1 <= q and q <= no then 
P := Appendd(P, Array([n, q, lcm(u, LtG[q]), str union {o}, SPolynomial(g, GO[q], t), m, u, LtG[q]])); 
elif no + 1 <= q and q <= 2*no then 
P := Appendd(P, Array([n, q - no, lcm(u, LtG[q]), str union {w}, SPolynomial(g, GW[q - no], t), m, u, LtG[q]])); 
else 
P := Appendd(P, Array([n, q - 2*no, lcm(u, LtG[q]), str union {b}, SPolynomial(g, B[q - 2*no], t), m, u, LtG[q]])); 
end if; 
pount := pount + 1; 
end do; 
RETURN(P); 
end proc;

Pairw := proc(g, f, t) 
local P, A, A1, a, i, j, u, na, q, C, AA; 
global pount, pk, TT, no, G, LtG, n, GW, GO, ps; 
P := Array([]); 
u := LeadingMonomial(g, t); 
na := ArrayNumElems(LtG); 
A := Array([seq(lcm(LtG[j], u)/u, j = 1 .. na)]); 
AA := Removee(A, n + no); 
A1 := Array([op({op(Basis(convert(AA, list), t))} minus {seq(LtG[j], j = 1 .. na)})]); 
pk := pk + na - ArrayNumElems(A1); 
for a in A1 do member(a, A, 'q'); 
if 1 <= q and q <= no then 
if n <> q or n = q and TT[n] = false then 
P := Appendd(P, Array([n, q, lcm(u, LtG[q]), {o, w}, SPolynomial(g, GO[q], t), n, u, LtG[q]])); 
else 
ps := ps + 1; 
end if; 
elif no + 1 <= q and q < n then 
if TT[n] = false or TT[q - no] = false then 
P := Appendd(P, Array([n, q - no, lcm(u, LtG[q]), {w}, SPolynomial(g, GW[q - no], t), n, u, LtG[q]])); 
else 
ps := ps + 1; 
end if; 
elif q = na then if 
TT[n] = false then 
P := Appendd(P, Array([n, infinity, lcm(u, LtG[q]), {w, f}, SPolynomial(g, f, t), n, u, LtG[q]])); 
else 
ps := ps + 1; 
end if; 
elif 2*no < q then 
P := Appendd(P, Array([n, q - 2*no, lcm(u, LtG[q]), {b, w}, SPolynomial(g, B[q - 2*no], t), n, u, LtG[q]])); 
end if; 
pount := pount + 1; 
end do; 
RETURN(P); 
end proc;

Pairc := proc(g, t) 
local P, A, A1, a, i, j, u, n, C, T, qq, ll; 
global pount, pk, B, LtB, G, LtG; 
u := w*mul(x[i], i = 1 .. 10); 
n := ArrayNumElems(LtG); 
A := Array([seq(lcm(LtG[j], u)/u, j = 1 .. n)]); 
A1 := Array([op({op(Basis(convert(A, list), t))} minus {seq(LtG[j], j = 1 .. n)})]); 
pk := pk + n - ArrayNumElems(A1); 
T := NULL; 
for a in A1 do 
member(a, A, 'q'); 
T := T, q; 
end do;
T := {seq(i, i = 1 .. n)} minus {T}; 
T := convert(T, list); 
T := Reverse(T); 
G := convert(G, list); 
LtG := convert(LtG, list); 
for qq to nops(T) do 
B := Appendd(B, G[T[qq]]); 
LtB := Appendd(LtB, LtG[T[qq]]); 
G := subsop(T[qq] = NULL, G); 
LtG := subsop(T[qq] = NULL, LtG); 
end do; 
G := convert(G, Array); 
LtG := convert(LtG, Array); 
ll := ArrayNumElems(G); 
P := Array([seq(Array([infinity, i, lcm(u, LtG[i]), {f, o}]), i = 1 .. ll)]); 
pount := pount + ll; 
RETURN(P); 
end proc;

CriterionS := proc(p, t) 
global S; 
if p[4] = {o, w} and TT[p[1]] = true and TT[p[2]] = true then 
if member([p[2], p[1], p[4]], S) or member([p[1], p[2], p[4]], S) then RETURN(false); 
end if; 
end if; 
if p[4] = {o} then 
if member([p[1], p[2], {w}], S) or member([p[2], p[1], {w}], S) then RETURN(false); 
end if; 
end if; 
if p[4] = {w} then 
if member([p[1], p[2], {o}], S) or member([p[2], p[1], {o}], S) then RETURN(false); 
end if; 
end if; 
RETURN(true); 
end proc;

CommonFactor := proc(f) 
local A, gc, i, g, G; 
if f = 0 then 
RETURN(0); 
else 
A := [op(expand(f))]; 
gc := A[1]; 
for i from 2 to nops(A) do 
gc := gcd(gc, A[i]); 
end do; 
RETURN(simplify(f/gc)); 
end if; 
end proc;

BSH := proc(F, t) 
local G, LtG, P, i, p, s, h, g, n, M, m, LtM, f; 
global count, pount, rount; 
G := Array([seq(CommonFactor(f), f in F)]); 
LtG := Array([seq(LeadingMonomial(g, t), g in G)]); 
P := Array([]); 
for i from 2 to ArrayNumElems(G) do 
P := Extendd(P, Pair(G[1 .. i - 1], LtG[1 .. i - 1], G[i], t)); 
end do; 
while ArrayNumElems(P) <> 0 do 
P := Sort(P, t); 
p := P[-1]; 
P := Array(1 .. ArrayNumElems(P) - 1, P); 
s := SPolynomial(p[1], p[2], t); 
count := count + 1; 
h := CommonFactor(Division(CommonFactor(s), G, t)); 
if h <> 0 then 
P := Extendd(P, Pair(G, LtG, h, t)); 
G := Appendd(G, h); 
LtG := Appendd(LtG, LeadingMonomial(h, t)); 
else 
rount := rount + 1; 
end if; 
end do; 
LtM := Array(Basis(convert(LtG, list), t)); 
M := Array([]); 
for m in LtM do member(m, LtG, 'q'); 
M := Appendd(M, G[q]); 
end do; 
RETURN(M, LtM); 
end proc;

BSHSat := proc(F, f, t) #This algorithm computes a Grobner basis for an ideal using the new structure described in our paper.
local P, i, p, s, h, g, hw, sw, cp, LtGO, LtGW, Lth, Lthw, p23, A, pw, firsttime, firstbytes, secondtime, secondbytes; 
global count, pount, rount, pk, TT, B, LtB, G, LtG, n, str, no, nb, GW, GO, LtGG, m, S, ps, Ltf; 
firsttime, firstbytes := kernelopts(cputime, bytesused); 
count := 0; pount := 0; rount := 0; pk := 0; m := 0; ps := 0; 
G, LtG := BSH(F, t); 
B := Array([]); 
LtB := Array([]); 
LtGG := Array([]); 
Ltf := LeadingMonomial(f, t); 
A := Array([]); 
A := Extendd(A, Pairc(f, t)); 
GO, LtGO := G, LtG; 
GW, LtGW := Array([]), Array([]); 
TT := Array([]); 
nb := ArrayNumElems(B); 
no := ArrayNumElems(GO); 
cp := 0; 
while ArrayNumElems(A) <> 0 do 
p := A[1]; 
A := Removee(A, 1); 
s := CommonFactor(SPolynomial(f, GO[p[2] - cp], t)); 
count := count + 1; 
h := CommonFactor(Division(s, Extendd(Extendd(GO, GW), Appendd(B, f)), t)); if h <> 0 then 
Lth := LeadingMonomial(h, t); 
if Lth = LeadingMonomial(s, t) then 
TT := Appendd(TT, true); 
else 
TT := Appendd(TT, false); 
end if; 
GW := Appendd(GW, h); 
LtGW := Appendd(LtGW, Lth); 
LtGG := Appendd(LtGG, Lth); 
else 
B := Appendd(B, GO[p[2] - cp]); 
nb := nb + 1; 
LtB := Appendd(LtB, LtGO[p[2] - cp]); 
GO := Removee(GO, p[2] - cp); 
no := no - 1; 
LtGO := Removee(LtGO, p[2] - cp); 
cp := cp + 1; 
end if; 
end do; 
P := Array([]); 
G := Extendd(Extendd(GO, GW), Appendd(B, f)); 
LtG := Extendd(Extendd(LtGO, LtGW), Appendd(LtB, Ltf)); 
for i to no do 
n := i; 
P := Extendd(P, Pairw(GW[i], f, t)); 
end do; 
S := Array([]); 
G := Extendd(GO, GW); 
LtG := Extendd(LtGO, LtGW); 
m := no; 
while ArrayNumElems(P) <> 0 do 
P := Sort(P, t); 
p := P[-1]; 
P := Array(1 .. ArrayNumElems(P) - 1, P); 
if nops(remove(has, [p[7], p[8]], w)) = 2 and gcd(p[7], p[8]) = 1 then
ps := ps + 1; 
else 
if CriterionS(p, t) = true then 
S := Append(S, [p[1], p[2], p[4]]); 
s := CommonFactor(p[5]); 
count := count + 1; 
h := CommonFactor(Division(s, Extendd(G, Appendd(B, f)), t)); 
if h <> 0 then 
Lth := LeadingMonomial(h, t); 
m := m + 1; 
LtGG := Appendd(LtGG, Lth); 
G, LtG := Extendd(Extendd(GO, GW), B), Extendd(Extendd(LtGO, LtGW), LtB);
sw := CommonFactor(SPolynomial(h, f, t)); 
hw := CommonFactor(Division(sw, Extendd(G, Appendd(B, f)), t)); 
if hw <> 0 then 
str := {o}; 
n := no + 1; 
P := Extendd(P, PairS(h, t)); 
Lthw := LeadingMonomial(hw, t); 
m := m + 1; 
LtGG := Appendd(LtGG, Lthw); 
str := {w}; 
P := Extendd(P, PairS(hw, t)); 
GO := Appendd(GO, h); 
LtGO := Appendd(LtGO, Lth); 
GW := Appendd(GW, hw); 
LtGW := Appendd(LtGW, Lthw); 
G := Extendd(GO, GW); 
LtG := Extendd(LtGO, LtGW); 
no := no + 1; 
if Lthw = LeadingMonomial(sw, t) then 
TT := Appendd(TT, true); 
else 
TT := Appendd(TT, false); 
end if; 
else 
str := {b}; 
n := nb + 1; 
P := Extendd(P, PairS(h, t)); 
B := Appendd(B, h); 
LtB := Appendd(LtB, Lth); 
nb := nb + 1; 
rount := rount + 1; 
end if; 
else 
rount := rount + 1; 
end if; 
else 
ps := ps + 1; 
end if; 
end if; 
end do; 
secondtime, secondbytes := kernelopts(cputime, bytesused); 
printf("%-1s %1s %1s %1s:         %3a %3a\n", The, cpu, time, is, secondtime - firsttime, sec); 
printf("%-1s %1s %1s:         %3a %3a\n", The, used, memory, secondbytes - firstbytes, bytes); 
printf("%-1s %1s %1s %1s:         %3a\n", number, of, Spoly, is, count);
printf("%-1s %1s %1s %1s:         %3a\n", number, of, pairs, is, pount);
printf("%-1s %1s %1s %1s:         %3a\n", number, of, zeros, is, rount);
printf("%-1s %1s %1s:           %3a\n", BSH, criterians, is, pk);
printf("%-1s %1s %1s:           %3a\n", new, criterian, is, ps);
RETURN(Extendd(G, B)); 
end proc;

BSHC := proc(F, t)#This is the Berkesch-Schreyer algorithm that computes a Grobner basis for the saturation of the given ideal (by using Rabinowitsch's trick and by removing the common factors).  
local G, LtG, P, i, p, s, h, g, f, n, firsttime, firstbytes, secondtime, secondbytes; 
global count, pount, rount, pk; 
firsttime, firstbytes := kernelopts(cputime, bytesused); 
pk := 0; count := 0; pount := 0; rount := 0; 
G := Array([seq(CommonFactor(f), f in F)]); 
LtG := Array([seq(LeadingMonomial(g, t), g in G)]); 
P := Array([]); 
for i from 2 to ArrayNumElems(G) do 
P := Extendd(P, Pair(G[1 .. i - 1], LtG[1 .. i - 1], G[i], t)); 
end do; 
while ArrayNumElems(P) <> 0 do 
P := Sort(P, t); 
p := P[-1]; 
P := Array(1 .. ArrayNumElems(P) - 1, P); 
s := SPolynomial(p[1], p[2], t); 
count := count + 1; 
h := CommonFactor(Division(CommonFactor(s), G, t)); 
if h <> 0 then 
P := Extendd(P, Pair(G, LtG, h, t)); 
G := Appendd(G, h); 
LtG := Appendd(LtG, LeadingMonomial(h, t)); 
else 
rount := rount + 1; 
end if; 
end do; 
secondtime, secondbytes := kernelopts(cputime, bytesused); 
printf("%-1s %1s %1s %1s:         %3a %3a\n", The, cpu, time, is, secondtime - firsttime, sec); 
printf("%-1s %1s %1s:         %3a %3a\n", The, used, memory, secondbytes - firstbytes, bytes); 
printf("%-1s %1s %1s %1s:         %3a\n", number, of, Spoly, is, count);
printf("%-1s %1s %1s %1s:         %3a\n", number, of, pairs, is, pount); 
printf("%-1s %1s %1s %1s:         %3a\n", number, of, zeros, is, rount);
printf("%-1s %1s %1s:           %3a\n", BSH, criterians, is, pk);
RETURN(G); 
end proc;

tord := plex(w, seq(x[i], i = 1 .. 10));

F := [x[1]^5*x[2]^3*x[3]*x[4]*x[5]^3*x[6]*x[7]^5*x[8]^4*x[10]^2 - x[1]*x[2]^3*x[3]*x[4]^5*x[5]^3*x[6]^2*x[7]^5*x[8]^2*x[9]*x[10]^2, x[1]^5*x[2]^4*x[3]^4*x[4]^3*x[6]^3*x[7]*x[8]*x[9]^4*x[10] - x[1]^4*x[2]^3*x[7]*x[8]^4*x[9]^5*x[10]^3, -x[1]^4*x[2]^5*x[3]^3*x[4]^2*x[5]^4*x[6]^4*x[8]*x[9]*x[10]^3 - x[1]^2*x[3]^2*x[5]^4*x[7]^3*x[8]*x[9]^2*x[10]];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);

F := [x[1]^4*x[3]^3*x[4]^4*x[5]^2*x[6]^3*x[7]^3*x[8]^4*x[9]^4*x[10] + x[1]^4*x[2]^5*x[3]^3*x[5]^2*x[6]^3*x[7]^2*x[8]^3*x[9]^5, -x[1]^4*x[2]^3*x[4]^2*x[5]^4*x[6]^4*x[9]^4*x[10]^4 + x[1]^3*x[2]^4*x[5]^2*x[6]^2*x[8]^4*x[9]^2, -x[1]^4*x[2]^5*x[4]^3*x[5]^5*x[6]*x[8]^2*x[9]^2*x[10]^4 + x[1]^2*x[2]*x[3]^3*x[4]^2*x[9]^3*x[10], -x[1]^5*x[3]^2*x[4]*x[5]^2*x[6]^4*x[7]^5*x[9]^2*x[10]^2 - x[1]^2*x[2]^4*x[3]*x[5]*x[6]*x[7]^5*x[8]*x[9]^2*x[10]^5];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);


F := [x[1]^4*x[2]^5*x[3]^5*x[4]^4*x[5]^3*x[6]^2*x[7]^2*x[8]*x[9]*x[10]^3 + x[2]*x[3]^4*x[4]^4*x[5]*x[6]^4*x[7]^5*x[8]^2*x[9]^3, x[1]^3*x[3]^4*x[4]^5*x[6]^3*x[7]^4*x[8]^5*x[10]^4 + x[2]^3*x[3]*x[4]^5*x[5]^3*x[6]^4*x[7]^5*x[8]*x[9]^5, x[3]^4*x[4]^5*x[5]*x[6]^4*x[7]^5*x[9]^2*x[10]^2 + x[2]^4*x[3]^4*x[4]^2*x[7]*x[8]^2*x[9]*x[10], x[1]^4*x[2]*x[4]^2*x[5]^5*x[6]^4*x[7]^3*x[8]^4*x[9]*x[10] + x[1]^3*x[3]^3*x[4]^5*x[5]^3*x[6]*x[7]*x[9]^4*x[10]];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);


F := [-x[1]*x[2]^4*x[3]^5*x[4]*x[5]^2*x[6]^3*x[7]^3*x[8]^5*x[9]^4*x[10]^2 - x[1]^4*x[2]*x[3]^2*x[4]*x[5]*x[6]^3*x[7]^2*x[8]^2*x[10], x[2]^5*x[3]^4*x[4]^5*x[5]^3*x[6]^2*x[7]^5*x[8]^4*x[9]^2*x[10]^4 - x[1]^3*x[2]^4*x[3]^5*x[4]*x[5]^2*x[6]*x[7]^5*x[8]^3*x[10]^4, x[1]*x[2]^3*x[3]*x[5]^4*x[7]^3*x[8]^3*x[9]^4*x[10]^2 + x[1]*x[2]^2*x[5]^2*x[6]^4*x[7]^4*x[8]*x[9]*x[10], x[1]*x[2]^2*x[3]^4*x[4]^4*x[5]^3*x[6]^2*x[7]^2*x[8]^4*x[9]^5*x[10]^2 - x[2]^3*x[3]^2*x[4]^4*x[5]^3*x[6]^5*x[7]^3*x[8]^3*x[9]*x[10], -x[1]^4*x[2]*x[3]^3*x[4]^4*x[5]^3*x[6]^5*x[8]^5*x[9]^2 + x[1]*x[2]^3*x[4]^4*x[5]*x[6]^2*x[7]*x[8]^3*x[9]^5*x[10]^5];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);


F := [-x[1]^10*x[2]^10*x[3]*x[4]^9*x[6]^8*x[7]^3*x[8]^9*x[10]^6 + x[1]^3*x[2]^10*x[3]^9*x[4]^6*x[5]^9*x[6]^7*x[7]^3*x[8]*x[9]^3*x[10]^4, -x[2]^6*x[3]^5*x[5]^3*x[6]^9*x[7]^10*x[8]^6*x[10]^8 + x[1]^3*x[2]^2*x[3]^4*x[5]^5*x[6]^10*x[7]^2*x[8]^8*x[9]^6*x[10]^5, -x[1]^6*x[2]^9*x[3]^6*x[4]^2*x[6]^2*x[7]^9*x[8]^6*x[9]^9*x[10]^6 - x[1]^6*x[2]^9*x[3]*x[4]*x[5]^5*x[7]^5*x[8]^4*x[9]^2*x[10]^10];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);


F := [x[1]^2*x[2]*x[3]^8*x[4]^6*x[5]^9*x[6]^10*x[7]^6*x[8]^8*x[9]^7*x[10]^10 + x[1]*x[2]^9*x[3]*x[4]*x[5]^3*x[6]^2*x[7]^10*x[8]^4*x[10]^6, x[1]^5*x[3]^4*x[4]^8*x[5]*x[6]^6*x[7]^3*x[8]^4*x[9]^9*x[10]^9 + x[1]^2*x[2]^8*x[3]^10*x[4]^10*x[5]^3*x[6]*x[8]^9*x[9]^3, x[1]^6*x[2]^9*x[3]^5*x[4]^5*x[5]^8*x[6]^9*x[7]^10*x[8]^3*x[10]^4 - x[2]^4*x[3]^6*x[4]^9*x[5]^7*x[6]^9*x[7]^8*x[8]^3*x[9]^8*x[10]^4];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);


F := [-x[1]^9*x[2]^10*x[3]*x[4]*x[5]^3*x[6]^7*x[7]^10*x[8]^2*x[9]^8*x[10]^9 - x[1]^6*x[2]^9*x[3]^5*x[4]*x[5]^10*x[6]^3*x[7]^5*x[8]^4*x[9]^10, -x[1]*x[2]^10*x[3]^7*x[4]^9*x[5]^8*x[6]^10*x[8]^5*x[9]^6*x[10]^6 + x[1]^2*x[2]^4*x[3]^2*x[4]^7*x[5]^9*x[6]^2*x[9]^2*x[10]^4, -x[1]^6*x[2]^3*x[3]^2*x[4]^9*x[5]^8*x[6]^9*x[7]^10*x[8]^3*x[9]^8*x[10]^6 - x[1]^9*x[2]^9*x[3]^9*x[4]^10*x[5]*x[6]^7*x[7]^3*x[8]*x[9]^8*x[10]^4, -x[1]^6*x[2]*x[3]^8*x[4]^4*x[5]^8*x[6]^8*x[7]^3*x[8]^2*x[9]^3*x[10]^5 - x[2]^9*x[3]*x[4]^10*x[5]^3*x[6]^6*x[7]^7*x[9]^5*x[10]^3];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);


F := [-x[1]^10*x[2]^6*x[3]*x[5]^4*x[6]^9*x[7]^2*x[8]^9*x[9]^3*x[10]^5 + x[1]^6*x[2]^8*x[3]^5*x[5]*x[6]^8*x[7]^4*x[8]^2*x[9]*x[10]^4, x[1]^8*x[2]^9*x[4]^4*x[5]^8*x[8]^2*x[9]^8*x[10]^8 - x[1]^2*x[2]^8*x[3]^7*x[4]^4*x[5]^3*x[6]^2*x[7]^2*x[8]^8*x[9]^2*x[10]^7, x[1]^10*x[2]^8*x[3]^3*x[4]^8*x[5]*x[6]^4*x[7]^4*x[8]^7*x[10] - x[1]^4*x[2]^7*x[3]*x[5]^8*x[6]^6*x[8]^8*x[9]^4, -x[1]^3*x[2]^7*x[3]^4*x[5]^4*x[6]^9*x[7]^4*x[8]^4*x[9]^4*x[10]^9 + x[1]^5*x[2]^5*x[3]^6*x[5]^3*x[6]*x[7]^7*x[8]*x[9]*x[10]^3]; 
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);

F := [x[1]^12*x[2]^7*x[3]^18*x[4]^3*x[5]^14*x[6]^19*x[7]^13*x[8]^20*x[9]^8*x[10]^3 - x[1]^6*x[2]^15*x[3]^3*x[4]^12*x[5]^2*x[6]^4*x[7]^2*x[8]^9*x[9]^8*x[10]^15, x[1]^16*x[2]^19*x[3]^16*x[4]^18*x[5]^15*x[6]^8*x[7]^15*x[8]^9*x[9]^18*x[10]^2 + x[1]^4*x[2]^20*x[3]^19*x[4]^7*x[5]^19*x[6]^5*x[7]^3*x[8]^9*x[9]^12*x[10]^3];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);
        
F := [x[1]^20*x[2]^11*x[3]^18*x[4]^30*x[5]^7*x[6]^25*x[7]^2*x[8]^16*x[9]^16*x[10]^2 - x[1]^4*x[2]^9*x[3]^25*x[4]^9*x[5]^26*x[6]*x[7]^7*x[8]^19*x[9]^14*x[10], x[1]^21*x[2]^7*x[3]^25*x[4]^28*x[5]*x[6]^10*x[7]^10*x[8]^29*x[9]^5*x[10]^7 + x[1]^22*x[2]^8*x[3]^19*x[4]^25*x[5]^2*x[6]^17*x[7]^24*x[8]^16*x[10]^4];
BSHSat(F,mul(x[i], i = 1 .. 10),tord);
BSHC(F,tord);