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TestIdeals :: frobeniusPower

frobeniusPower -- computes the (generalized) Frobenius power of an ideal

Synopsis

Description

frobeniusPower(t,I) computes the generalized Frobenius power I[t], as introduced by Hernandez, Teixeira, and Witt. If the exponent is a power of the characteristic, this is just the usual Frobenius power:

i1 : R = ZZ/5[x,y];
i2 : I = ideal(x,y);

o2 : Ideal of R
i3 : frobeniusPower(125,I)

             125   125
o3 = ideal (x   , y   )

o3 : Ideal of R

If n is an arbitrary nonnegative integer, then write the base p expansion of n as follows: n = a0 + a1 p + a2 p2 + ... + ar pr. Then the nth Frobenius power of I is defined as follows: I[n] = (Ia0)(Ia1)[p](Ia2)[p2]…(Iar)[pr].

i4 : R = ZZ/3[x,y];
i5 : I = ideal(x,y);

o5 : Ideal of R
i6 : adicExpansion(3,17)

o6 = {2, 2, 1}

o6 : List
i7 : J1 = I^2*frobenius(1,I^2)*frobenius(2,I);

o7 : Ideal of R
i8 : J2 = frobeniusPower(17,I);

o8 : Ideal of R
i9 : J1 == J2

o9 = true

If t is a rational number of the form t = a/pe, then I[t] = (I[a])[1/pe].

i10 : R = ZZ/5[x,y,z];
i11 : I = ideal(x^50*z^95, y^100+z^27);

o11 : Ideal of R
i12 : frobeniusPower(4/5^2,I)

              4   4 3   8 2   12    16
o12 = ideal (z , y z , y z , y  z, y  )

o12 : Ideal of R
i13 : frobeniusRoot(2,frobeniusPower(4,I))

              4   4 3   8 2   12    16
o13 = ideal (z , y z , y z , y  z, y  )

o13 : Ideal of R

If t is an arbitrary nonegative rational number, and {tn }= {an/pen } is a sequence of rational numbers converging to t from above, then I[t] is the largest ideal in the increasing chain of ideals {I[tn] }.

i14 : p = 7;
i15 : R = ZZ/p[x,y];
i16 : I = ideal(x^50,y^30);

o16 : Ideal of R
i17 : t = 6/19;
i18 : expon = e -> ceiling( p^e*t )/p^e; -- a sequence converging to t from above
i19 : scan( 5, i -> print frobeniusPower(expon(i),I) )
        50   30
ideal (x  , y  )
        12   7 8   14 4   21
ideal (y  , x y , x  y , x  )
        9   2 8   7 5   8 4   14    15
ideal (y , x y , x y , x y , x  y, x  )
        9     8   7 4   14
ideal (y , x*y , x y , x  )
        9     8   7 4   14
ideal (y , x*y , x y , x  )
i20 : frobeniusPower(t,I)

              9     8   7 4   14
o20 = ideal (y , x*y , x y , x  )

o20 : Ideal of R

See also

Ways to use frobeniusPower :