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


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 :

For the programmer

The object frobeniusPreimage is a method function with options.