# Alternative algorithm to compute the symbolic powers of a prime ideal in positive characteristic

Given a prime ideal P in a regular ring of positive characteristic, symbPowerPrimePosChar computes its symbolic powers. Unfortunately, this algorithm is slower than others.

 i1 : R = ZZ/7[x,y,z] o1 = R o1 : PolynomialRing i2 : P = ker map(ZZ/7[t],R,{t^3,t^4,t^5}) 2 2 2 3 o2 = ideal (y - x*z, x y - z , x - y*z) o2 : Ideal of R i3 : J = symbPowerPrimePosChar(P,2) 4 2 2 2 2 3 3 2 2 3 3 2 4 3 o3 = ideal (y - 2x*y z + x z , x y - x y*z - y z + x*z , x y - x z - y z ------------------------------------------------------------------------ 2 5 3 2 3 + x*y*z , x + x*y - 3x y*z + z ) o3 : Ideal of R

The symbolic powers of P do not coincide with its powers.

 i4 : J == P^2 o4 = false

We can also test it a bit faster, without computing the symbolic powers of $P$.

 i5 : isSymbolicEqualOrdinary(P,2) o5 = false