A={{1,1,1,1},{1,5,10,25}}
R=QQ[p,n,d,q, Degrees => transpose A]
degree d
degree q
degree(p^4*n^8*d^10*q^3)
h=basis({20,135},R)
rank source h
h
rank source basis({100,1000},R)


S=QQ[x,y,d,p,n,q, MonomialOrder=>Lex, MonomialSize =>16]
I=ideal(p-x*y, n-x*y^5, d-x*y^10, q-x*y^25)
transpose gens gb I
Sq=S/I
-- express one dollar in 10 coins
x^10*y^100
-- we don't need any dime to express 10 dollars with 100 coins
x^100*y^1000
-- we cannot express 10 dollars with less then forty coins
x^39*y^1000


weight=(5,7,13,17)
T=QQ[x,y,p,n,d,q, Weights=> {{1,1,0,0,0,0}, {0,0,weight}},MonomialSize =>16]
describe T
I=ideal(p-x*y, n-x*y^5, d-x*y^10, q-x*y^25)
Tq=T/I
-- express one dollar in 10 coins
x^10*y^100
-- we don't need any dime to express 10 dollars with 100 coins
x^100*y^1000
optimal solution which involves all four coins
x^234*y^5677


load "LLL.m2"

oneBinomial = (R,C,m)-> (
     pos:=1_R;
     neg:=1_R;
     scan(numgens R, i->( a:=C_(i,m); 
                          if a>0 then 
                             pos=pos*R_i^a 
                          else
                             neg=neg*R_i^(-a)));
     pos-neg)

R=QQ[x_1..x_5, Weights=>{1,5,5,1,0}] 
A=matrix{{1,1,1,1,1},{0,1,2,1,0},{0,0,1,2,1}}
B=matrix LLL syz A
I= ideal apply(rank source B, i->(oneBinomial(R,B,i)))
transpose gens gb I

S=R/I
x_1*x_2^(10)*x_3^(10)*x_4^4