# Frobenius liftings of du Val singularities.

## 2246 days ago by mz248366

# E_6 & E_8 p = 11; R.<x,y,z> = PolynomialRing(GF(p),3,'xyz'); f = x^2+y^3+z^5; a = ceil(p/2); b = ceil((2*p)/3); c = ceil((4*p)/5); I = (f,x^p,y^(2*p),z^(4*p))*R; print I; g = 0; for i in range(0,p+1): for j in range(0,p-i+1): k = p - i - j; if i*j != 0 or i*k != 0 or j*k != 0: popr = (binomial(p,i)*binomial(p-i,j)); c = popr//p; # print "Work for: ",i,j,k," binomial: ",popr,c # print x^(2*i)*y^(3*j)*z^(5*k) in I; g = g + c*x^(2*i)*y^(3*j)*z^(5*k); else: # print "Don't work for: ",i,j,k continue # print g; g in I; #list(g)
 ```Ideal (z^5 + y^3 + x^2, x^11, y^22, z^44) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 11```
# Does the Frobenius lift to x^n + y^n + z^n - cones p = 17; n = 4; R.<x,y,z> = PolynomialRing(GF(p),3,'xyz'); f = x^n+y^n+z^n; I = (f,x^((n-1)*p),y^((n-1)*p),z^((n-1)*p))*R; print I; g = 0; for i in range(0,p+1): for j in range(0,p-i+1): k = p - i - j; if i*j != 0 or i*k != 0 or j*k != 0: popr = (binomial(p,i)*binomial(p-i,j)); c = popr//p; # print "Work for: ",i,j,k," binomial: ",popr,c # print x^(n*i)*y^(n*j)*z^(n*k) in I; g = g + c*x^(n*i)*y^(n*j)*z^(n*k); else: # print "Don't work for: ",i,j,k continue # print "Element =",g; print "Redukcja =",I.reduce(g); g in I;
 ```Ideal (x^4 + y^4 + z^4, x^51, y^51, z^51) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 17 Redukcja = y^32*z^36 + 4*y^28*z^40 + 3*y^24*z^44 - 5*y^20*z^48 False```
# D_{n+1} p = 61; n = 70; R.<x,y,z> = PolynomialRing(GF(p),3,'xyz'); f = x^2+z*y^2+z^n; I = (f,x^p,(y*z)^p,(y^2 + n*z^(n-1))^p)*R; print I; g = 0; for i in range(0,p+1): for j in range(0,p-i+1): k = p - i - j; if i*j != 0 or i*k != 0 or j*k != 0: popr = (binomial(p,i)*binomial(p-i,j)); c = popr//p; # print "Work for: ",i,j,k," binomial: ",popr,c g = g + c*x^(2*i)*(z*y^2)^j*z^(n*k) else: # print "Don't work for: ",i,j,k continue g in I;
 ```Ideal (z^70 + y^2*z + x^2, x^61, y^61*z^61, 9*z^4209 + y^122) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 61 True```
p = 3; R.<x,y,z> = PolynomialRing(GF(p),3,'xyz'); M = FreeModule(R,p^3); M; B_basis = [] # B_1_R itd. def make_P(f_1,f_2,f_3): g = 0 for i in range(0,p+1): for j in range(0,p-i+1): k = p - i - j; if i*j != 0 or i*k != 0 or j*k != 0: popr = (binomial(p,i)*binomial(p-i,j)); c = popr//p; g = g + c*(f_1)^i*(f_2)^j*(f_3)^k return g P = make_P(x^2,y^3,z^5) def in_basis(f): res = M.zero() for d,mon in zip(f.coefficients(), f.exponents()): # print d,mon (a,b,c) = mon res = res + d*(x^(a//p)*y^(b//p)*z^(c//p))*M.gen((a % p) + p*(b % p) + p^2*(c % p)) return res poly = 2*x^2 + y^3 + 2*x*y^4*z^5; print poly dfx = poly.derivative(x) dfy = poly.derivative(y) dfz = poly.derivative(z) print "(",dfx,")dx + (",dfy,")dy + (",dfz,")dz" elem = in_basis(poly); print elem for i in range(1,p^3): B_basis.append(M.gen(i))
 ```-x*y^4*z^5 + y^3 - x^2 ( -y^4*z^5 + x )dx + ( -x*y^3*z^5 )dy + ( x*y^4*z^4 )dz (y, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -y*z, 0, 0, 0, 0)```