Background:
Theorem (Saito): If R is a formal power series ring over a field of char 0, and f \in R is a power series with an isolated singularity, then f\in j(f), the Jacobian ideal iff f becomes quasi-homogeneous after a change of variables.
This can be tested over an affine ring by testing f % (j(f)+ideal vars S). If the result is 0 we call f crypto-quasi-homogeneous.
Theorem (Lejeune-Teisser; see Swanson-Huneke Thm 7.1.5) f \in integral closure(ideal apply(numgens R,i-> x_i*df/dx_i))
Question (Huneke): Is f actually contained in the maximal ideal times the integral closure of ideal apply(numgens R,i-> df/dx_i).
Note that the answer is trivially yes if f is crypto-quasi-homogeneous.
Huneke has shown that if the answer is always yes, then the Eisenbud-Mazur conjecture on evolutions is true.
i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing |
i2 : f = random(3,R)+random(4,R)+random(5,R)
3 5 4 3 2 2 3 4 4 5 5 3 4 10 3 2 2
o2 = --x + x y + 5x y + 10x y + -x*y + -y + -x z + --x y*z + 3x y z +
10 3 3 7 9
------------------------------------------------------------------------
7 3 7 4 1 3 2 2 2 5 2 2 2 3 2 3 2 3 3
-x*y z + -y z + -x z + 3x y*z + -x*y z + -y z + -x z + 5x*y*z +
8 2 2 6 5 2
------------------------------------------------------------------------
6 2 3 2 4 5 4 5 5 7 4 1 3 7 2 2 5 3 4 7 3
-y z + -x*z + -y*z + -z + --x + -x y + -x y + -x*y + 2y + --x z
5 5 4 7 10 2 3 2 10
------------------------------------------------------------------------
2 6 2 3 3 2 2 2 2 5 2 2 3 2 3 4
+ 7x y*z + -x*y z + 6y z + -x z + -x*y*z + -y z + x*z + -y*z + 5z
7 7 3 4 9
------------------------------------------------------------------------
9 3 1 2 1 2 3 3 9 2 3 2 3 2 7 2 7 3
+ -x + -x y + -x*y + -y + -x z + x*y*z + -y z + -x*z + -y*z + -z
2 2 2 2 4 4 4 4 9
o2 : R
|
i3 : testHunekeQuestion f power series is crypto-quasi-homogeneous o3 = yes |
The function y^4-2*x^3*y^2-4*x^5*y+x^6-x^7 is defines the simplest plane curve singularity with 2 characteristic pairs, and is thus NOT crypto- quasi-homogeneous.
i4 : R = QQ[x,y] o4 = R o4 : PolynomialRing |
i5 : f = (y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)
7 6 5 3 2 4
o5 = - x + x - 4x y - 2x y + y
o5 : R
|
i6 : testHunekeQuestion f power series is not crypto-quasi-homogeneous o6 = yes |
The object testHunekeQuestion is a method function.