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Divisor :: reflexivePower

reflexivePower -- computes a reflexive power of an ideal in a normal domain

Synopsis

Description

This function returns the n’th reflexive power of I. By definition this is the reflexification of In, or in other words, Hom(Hom(In, R), R).

i1 : R = QQ[x,y,z]/ideal(x^2-y*z);
i2 : J = ideal(x,y);

o2 : Ideal of R
i3 : reflexivePower(5, J)

             3     2
o3 = ideal (y , x*y )

o3 : Ideal of R
i4 : reflexivePower(6, J)

            3
o4 = ideal y

o4 : Ideal of R

This function is typically much faster than reflexifying In however. We can obtain this speedup because in a normal domain, the reflexification of In is the same as the reflexification of the ideal generated by the nth powers of the generators of I. Consider the example of a cone over a point on an elliptic curve.

i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
i6 : I = ideal(x-z,y-2*z);

o6 : Ideal of R
i7 : time J20a = reflexivePower(20, I);
     -- used 0.039965 seconds

o7 : Ideal of R
i8 : I20 = I^20;

o8 : Ideal of R
i9 : time J20b = reflexify(I20);
     -- used 0.336485 seconds

o9 : Ideal of R
i10 : J20a == J20b

o10 = true

See also

Ways to use reflexivePower :