potential -- Outputs the polynomial $f$ such that the dth power of the differentials is equal to $ f \cdot \text{id}$

Usage:

potential(C)
Inputs:
- C, a ZZ/d graded factorization, A ZZ/d-graded factorization
Outputs:
- a ring element,

Description

This method determines the potential of the factorization $C$ by first checking if $C$ a well-defined factorization using the isdFactorization function. If it is, it returns the potential $ f $; otherwise, it throws an error.

i1 : Q = ZZ/101[a..d]

o1 = Q

o1 : PolynomialRing

i2 : f = a^2+b^2+c^2+d^2

      2    2    2    2
o2 = a  + b  + c  + d

o2 : Q

i3 : C = linearMF(f)

      8      8      8
o3 = Q  <-- Q  <-- Q
                    
     0      1      0

o3 : ZZdFactorization

i4 : potential C

      2    2    2    2
o4 = a  + b  + c  + d

o4 : Q

i5 : S = ZZ/101[a,b]

o5 = S

o5 : PolynomialRing

i6 : D = randomTailMF(a^3 + b^3, 2, 6, 3)

      5      5      5
o6 = S  <-- S  <-- S
                    
     0      1      0

o6 : ZZdFactorization

i7 : D.dd

          5                                 5
o7 = 1 : S  <----------------------------- S  : 0
               {1} | -a2 b2 0   0   0  |
               {1} | 0   0  -a2 b2  ab |
               {1} | 0   0  -ab -a2 b2 |
               {1} | 0   0  b2  ab  a2 |
               {2} | b   a  0   0   0  |

          5                         5
     0 : S  <--------------------- S  : 1
               | -a 0  0  0 b2 |
               | b  0  0  0 a2 |
               | 0  -a 0  b 0  |
               | 0  b  -a 0 0  |
               | 0  0  b  a 0  |

o7 : ZZdFactorizationMap

i8 : D.dd^2

          5                                         5
o8 = 0 : S  <------------------------------------- S  : 0
               | a3+b3 0     0     0     0     |
               | 0     a3+b3 0     0     0     |
               | 0     0     a3+b3 0     0     |
               | 0     0     0     a3+b3 0     |
               | 0     0     0     0     a3+b3 |

          5                                             5
     1 : S  <----------------------------------------- S  : 1
               {1} | a3+b3 0     0     0     0     |
               {1} | 0     a3+b3 0     0     0     |
               {1} | 0     0     a3+b3 0     0     |
               {1} | 0     0     0     a3+b3 0     |
               {2} | 0     0     0     0     a3+b3 |

o8 : ZZdFactorizationMap

i9 : potential D

      3    3
o9 = a  + b

o9 : S

Ways to use potential:

potential(ZZdFactorization)

For the programmer

The object potential is a method function.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:5814:0.

potential -- Outputs the polynomial $f$ such that the dth power of the differentials is equal to $ f \cdot \text{id}$

Description

See also

Ways to use potential:

For the programmer