# frobeniusPreimage -- finds the ideal of elements mapped into a given ideal, under all $p^{-e}$-linear maps

## Description

Given an ideal $Q$ in a ring $R$, one frequently considers $I_e(Q)$. This is the ideal of elements $x \in R$ such that $\phi(x^{1/p^e}) \in Q$ for all $\phi : R^{1/p^e} \to R$. Sometimes this ideal is called the Frobenius pre-image. In a regular ring, it agrees with the frobenius power $Q^{[p^e]}$.

 i1 : R = ZZ/7[x,y,z]/ideal(x*y-z^2); i2 : Q = ideal(x, z); o2 : Ideal of R i3 : frobeniusPreimage(1, Q) 3 4 o3 = ideal (0, x z, x ) o3 : Ideal of R

In the previous example $I_1(Q)$ agrees with $Q^{(p)}$, the $p$th symbolic power of $Q$.

## Ways to use frobeniusPreimage :

• "frobeniusPreimage(ZZ,Ideal)"

## For the programmer

The object frobeniusPreimage is .