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

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)

     7 5   2 4    5 3 2    1 2 3   4   4   5 5   6 4    5 3      4 2 2   
o2 = -x  + -x y + -x y  + --x y  + -x*y  + -y  + -x z + -x y*z + -x y z +
     2     5      7       10       7       4     5      9        3       
     ------------------------------------------------------------------------
     5   3    2 4    5 3 2   3 2   2   5   2 2   8 3 2    9 2 3   6     3  
     -x*y z + -y z + -x z  + -x y*z  + -x*y z  + -y z  + --x z  + -x*y*z  +
     9        9      3       7         9         5       10       7        
     ------------------------------------------------------------------------
     9 2 3       4   2   4   9 5   6 4     3    2 2 2       3   7 4   5 3   
     -y z  + 6x*z  + -y*z  + -z  + -x  + 6x y + -x y  + 5x*y  + -y  + -x z +
     4               9       8     7            9               8     4     
     ------------------------------------------------------------------------
      3 2      10   2    5 3    3 2 2          2     2 2   3   3   2   3  
     --x y*z + --x*y z + -y z + -x z  + 10x*y*z  + 5y z  + -x*z  + -y*z  +
     10         9        6      7                          2       5      
     ------------------------------------------------------------------------
     5 4   9 3   9 2    7   2    7 3   3 2    7        7 2     7   2       2
     -z  + -x  + -x y + -x*y  + --y  + -x z + -x*y*z + -y z + --x*z  + 7y*z 
     3     2     4      4       10     4      9        3      10            
     ------------------------------------------------------------------------
       3 3
     + -z
       7

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:

  • testHunekeQuestion(RingElement)

For the programmer

The object testHunekeQuestion is a method function.

The source of this document is in IntegralClosure.m2:2082:0.