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testHunekeQuestion -- tests a conjecture on integral closures strengthening the Eisenbud-Mazur conjecture

Synopsis

Description

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

See also

Ways to use testHunekeQuestion :

For the programmer

The object testHunekeQuestion is a method function.