frobeniusRoot(e, I)
frobeniusRoot(e, M)
frobeniusRoot(e, A)
frobeniusRoot(e, expList, idealList)
frobeniusRoot(e, expList, idealList, I)
frobeniusRoot(e, m, I)
frobeniusRoot(e, a, f)
frobeniusRoot(e, a, f, I)
In a polynomial ring $R = k[x_1, \ldots, x_n]$ with coefficients in a field of positive characteristic $p$, the $p^e$-th Frobenius root $I^{[1/p^e]}$ of an ideal $I$ is the smallest ideal $J$ such that $I\subseteq J^{[p^e]}$ (= frobeniusPower(p^e, J)). Similarly, if $M$ is a submodule of $R^k$, the $p^e$-th Frobenius root of $M$, denoted $M^{[1/p^e]}$, is the smallest submodule $V$ of $R^k$ such that $M\subseteq V^{[p^e]}$. The function frobeniusRoot computes such ideals and submodules.
There are many ways to call frobeniusRoot. The simplest way is to call frobeniusRoot(e, I), which computes $I^{[1/p^e]}$.
|
|
|
The function frobeniusRoot works over arbitrary finite fields.
|
|
|
|
In the following example, for a submodule $M$ of $R^2$, frobeniusRoot(1, M) computes the smallest submodule $V$ of $R^2$ such that $M \subseteq V^{[2]}$.
|
|
|
|
For ease of use, one can also simply pass a matrix $A$ whose image is $M$, and frobeniusRoot(1, A) returns a matrix whose image is $V$.
|
Often, one wants to compute a Frobenius root of some product of powers of ideals, $I_1^{a_1}\cdots I_n^{a_n}$. This is best accomplished by calling frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}).
|
|
|
|
|
|
|
For legacy reasons, the last ideal in the list can be specified separately, using frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}, I). The last ideal, I, is just raised to the first power.
The following are additional ways of calling frobeniusRoot:
$\bullet$ frobeniusRoot(e, m, I) computes the $p^e$-th Frobenius root of the ideal $I^m$.
$\bullet$ frobeniusRoot(e, a, f) computes the $p^e$-th Frobenius root of the principal ideal ($f^{ a}$).
$\bullet$ frobeniusRoot(e, a, f, I) computes the $p^e$-th Frobenius root of the product $f^{ a}I$.
There are two valid inputs for the option FrobeniusRootStrategy, namely Substitution and MonomialBasis. In the computation of the $p^e$-th Frobenius root of an ideal $I$, each generator $f$ of $I$ is written in the form $f = \sum a_i^{p^e} m_i$, where each $m_i$ is a monomial whose exponents are less than $p^e$; then the collection of all the $a_i$, obtained for all generators of $I$, generates the Frobenius root $I^{[1/p^e]}$. Substitution and MonomialBasis use different methods for gathering these $a_i$, and sometimes one method is faster than the other.
The object frobeniusRoot is a method function with options.