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TestIdeals :: ascendIdeal

ascendIdeal -- finds the smallest phi-stable ideal containing a given ideal in a quotient of a polynomial ring.

Synopsis

Description

Let φ be the p-e linear map obtained by multiplying e-th Frobenius trace on a polynomial ring by h. Then this function finds the smallest φ-stable ideal containing J. The idea is to consider the ascending chain J, J+φ(J), J+φ(J)+φ2(J), .... We return the stable value. For instance, this can be used to compute the test ideal. Note if the ideal J is not an ideal in a polynomial ring, the function will do the computation with e-th Frobenius trace in the ambient polynomial ring, but will do the comparison inside the quotient ring (to see if we are done).

i1 : S = ZZ/5[x,y,z];
i2 : g = x^4+y^4+z^4;
i3 : h = g^4;
i4 : R = S/ideal(g);
i5 : ascendIdeal(1, h, ideal(y^3))

             2             2        2
o5 = ideal (z , y*z, x*z, y , x*y, x )

o5 : Ideal of R
i6 : ascendIdeal(1, h, ideal((sub(y, S))^3))

             2             2        2
o6 = ideal (z , y*z, x*z, y , x*y, x )

o6 : Ideal of S

The alternate ways to call the function allow the function to behave in a more efficient way. Indeed, frequently the polynomial passed is a power, ha. If a is large, we don’t want to compute ha; instead we try to keep the exponent small by only raising it to the minimal power needed to do computation at that time.

i7 : S = ZZ/5[x,y,z];
i8 : g = x^4+y^4+z^4;
i9 : R = S/ideal(g);
i10 : ascendIdeal(1, 4, g, ideal(y^3))

              2             2        2
o10 = ideal (z , y*z, x*z, y , x*y, x )

o10 : Ideal of R
i11 : ascendIdeal(1, 4, g, ideal((sub(y, S))^3))

              2             2        2
o11 = ideal (z , y*z, x*z, y , x*y, x )

o11 : Ideal of S

More generally, if h is a product of powers, h = h1a1…hnan, then you should pass ascendIdeal the lists expList={a1,...,an} and {h1,...,hn} of exponents and bases.

This method appared first in the work of Mordechai Katzman on star closure.

Ways to use ascendIdeal :