################################################################## # Construction of parameters with roots of prescribed valuations # ################################################################## # In this script, we compute examples of parameters d1,...,d6 in QQ[t] giving 36 positive roots with prescribed valuations. There are two different formulas, corresponding to two cones mapping to the two maximal cones in the Naruki fan. Each function takes 4 rational numbers corresponding to the scalars (a,b,c,d) of the extremal rays of each cone. # The functions encoded compute the valuations in t of the roots and check that they agree with the desired vector in the corresponding cone in Bergman(E6) associated to the given 4 scalars a,b,c,d. # We choose the cone [0,1,4,36] as the cone in the Bergman fan mapping to the cone aa2a3b we choose: a=0, a2=0+1, a3=0+1+4 and b=36. # We choose the cone [0,1,4,158] as the cone in the Bergman fan mapping to the cone aa2a3a4. We take a=0, a2=0+1, a3=0+1+4 and a4=0+1+158. ########################################################################### # Setting up the ambient rings with the W(E6) action. ########################################################################### # A list of variables d_1, ..., d_6 for the coefficients ds = [var("d%s"%i) for i in range(1,7)] # A list of variables E_1, ..., E_6 Es = [var("E%s"%i) for i in range(1,7)] # A list of variables F_12, ..., F_56 Fs = [var("F%s%s"%(i,j)) for i in range(1,7) for j in range(1,7) if i < j] # A list of variables G_1, ..., G_6 Gs = [var("G%s"%i) for i in range(1,7)] # A list of variables X_12, ..., X_65 Xs = [var("X%s%s"%(i,j)) for i in range(1,7) for j in range(1,7) if i != j] # A list of variables Y_123456, ..., Y_162534 Ys = [var("Y%s%s%s%s%s%s"%(i,j,k,l,m,n)) for i in range(1,7) for j in range(1,7) for k in range(1,7) for l in range(1,7) for m in range(1,7) for n in range(1,7) if i