rationalIntegralClosure -- computes integral closures over the rationals

Synopsis

Usage:

(fractions,relicR,icR,wticR) = rationalIntegralClosure(wtR,Rq,GB)
Inputs:
- wtR, weight matrix
- R0, polynomial ring over QQ with weight-over-grevlex order
- GB, set of generators for the ideal of relations
Outputs:
- fractions, a list, a list of numerators for a P-module generating set, the first doubling as a common denominator, hence a conductor element as well
- relicR, a list, a Gr"obner basis for the ideal of induced relations
- icR, a ring, a grevlex-over-weight polynomial ring over QQ for the presentation
- wticR, a matrix, weight matrix for the integral closure

Description

i1 : wtR = matrix{{11,6}};

              1       2
o1 : Matrix ZZ  <-- ZZ

i2 : R0 = QQ[y,x,Weights=> entries weightGrevlex(wtR)];

i3 : GB = {(y^2-3/4*y-15/17*x)^3-9*y*x^4*(y^2-3/4*y-15/17*x)-27*x^11};

i4 : (fractions,relicR,icR,wticR) = rationalIntegralClosure(wtR,R0,GB)
 9
x
(3, 1)
     11
(5, x  )
     9
(7, x )
7
{p_0^2-3*p_0+p_1*p_5^3+p_2*p_5^5+3*p_2*p_5^2+2*p_3*p_5^5-2*p_4*p_5^4+2*p_4*p_5^3+p_4*p_5+2*p_5^5, p_0*p_1-2*p_0*p_5^2-3*p_1-3*p_2*p_5-p_3*p_5^4+p_4*p_5^6-2*p_5^5-p_5^4, p_0*p_2+p_1*p_5^4+3*p_1*p_5-3*p_2+2*p_4*p_5^4+3*p_4*p_5+2*p_5^6, p_0*p_3-2*p_1*p_5+2*p_2-2*p_4*p_5+p_5^6, p_0*p_4-2*p_2*p_5+p_3*p_5^4-3*p_4+p_5^4, p_1^2+3*p_0*p_5-2*p_1*p_5^2+2*p_2*p_5+3*p_2-2*p_4*p_5^2+p_5^7, p_1*p_2-3*p_1-2*p_2*p_5^2+p_3*p_5^5-3*p_4*p_5-3*p_4-p_5^5, p_1*p_3+2*p_1-p_2*p_5^2+2*p_4*p_5+2*p_4, p_1*p_4-p_0*p_5+2*p_2, p_2^2-2*p_0*p_5+2*p_2+p_4*p_5^5, p_2*p_3-p_0*p_5+p_2, p_2*p_4-2*p_1-2*p_4+p_5^5, p_3^2+p_3-p_4*p_5^2+2*p_5, p_3*p_4-p_1-p_4, p_4^2-p_2}
      9
(13, x )
13
{p_0^2+2*p_0+4*p_1*p_5^3-p_2*p_5^5-3*p_2*p_5^2+4*p_3*p_5^5+6*p_4*p_5^4-6*p_4*p_5^3+5*p_4*p_5-3*p_5^5, p_0*p_1+4*p_0*p_5^2+2*p_1+6*p_2*p_5-4*p_3*p_5^4-p_4*p_5^6+6*p_5^5+3*p_5^4, p_0*p_2-p_1*p_5^4-3*p_1*p_5+2*p_2-5*p_4*p_5^4-p_4*p_5+4*p_5^6, p_0*p_3+4*p_1*p_5+6*p_2-3*p_4*p_5-p_5^6, p_0*p_4+4*p_2*p_5-p_3*p_5^4+2*p_4+4*p_5^4, p_1^2+p_0*p_5+4*p_1*p_5^2+6*p_2*p_5+p_2-3*p_4*p_5^2-p_5^7, p_1*p_2+6*p_1+4*p_2*p_5^2-p_3*p_5^5+2*p_4*p_5+2*p_4-4*p_5^5, p_1*p_3+5*p_1-p_2*p_5^2+6*p_4*p_5+6*p_4, p_1*p_4-p_0*p_5+5*p_2, p_2^2+4*p_0*p_5+3*p_2-p_4*p_5^5, p_2*p_3-p_0*p_5-4*p_2, p_2*p_4+4*p_1-3*p_4-p_5^5, p_3^2-4*p_3-p_4*p_5^2+6*p_5, p_3*p_4-p_1+4*p_4, p_4^2-p_2}
      9
(19, x )
19
{p_0^2+p_0+6*p_1*p_5^3-8*p_2*p_5^5-5*p_2*p_5^2+4*p_3*p_5^5+3*p_4*p_5^4-9*p_4*p_5^3+9*p_4*p_5-3*p_5^5, p_0*p_1-9*p_0*p_5^2+p_1-4*p_2*p_5-6*p_3*p_5^4-8*p_4*p_5^6+3*p_5^5-5*p_5^4, p_0*p_2-8*p_1*p_5^4-5*p_1*p_5+p_2-7*p_4*p_5^4-p_4*p_5+4*p_5^6, p_0*p_3-9*p_1*p_5-2*p_2+2*p_4*p_5-8*p_5^6, p_0*p_4-9*p_2*p_5-8*p_3*p_5^4+p_4+6*p_5^4, p_1^2-7*p_0*p_5-9*p_1*p_5^2-2*p_2*p_5-7*p_2+2*p_4*p_5^2-8*p_5^7, p_1*p_2-4*p_1-9*p_2*p_5^2-8*p_3*p_5^5+p_4*p_5+3*p_4-6*p_5^5, p_1*p_3+8*p_1-p_2*p_5^2-2*p_4*p_5-6*p_4, p_1*p_4-p_0*p_5+8*p_2, p_2^2-9*p_0*p_5-2*p_2-8*p_4*p_5^5, p_2*p_3-p_0*p_5+4*p_2, p_2*p_4-9*p_1+2*p_4-8*p_5^5, p_3^2+4*p_3-p_4*p_5^2-2*p_5, p_3*p_4-p_1-4*p_4, p_4^2-p_2}
      9
(23, x )
23
{p_0^2+11*p_0+3*p_1*p_5^3-4*p_2*p_5^5+11*p_2*p_5^2+10*p_3*p_5^5+10*p_4*p_5^4+7*p_4*p_5^3+7*p_4*p_5+4*p_5^5, p_0*p_1-9*p_0*p_5^2+11*p_1-2*p_2*p_5-3*p_3*p_5^4-4*p_4*p_5^6+10*p_5^5+8*p_5^4, p_0*p_2-4*p_1*p_5^4+11*p_1*p_5+11*p_2+6*p_4*p_5^4+9*p_4*p_5+10*p_5^6, p_0*p_3-9*p_1*p_5-9*p_2+p_4*p_5-4*p_5^6, p_0*p_4-9*p_2*p_5-4*p_3*p_5^4+11*p_4+3*p_5^4, p_1^2-8*p_0*p_5-9*p_1*p_5^2-9*p_2*p_5-8*p_2+p_4*p_5^2-4*p_5^7, p_1*p_2-2*p_1-9*p_2*p_5^2-4*p_3*p_5^5+11*p_4*p_5-10*p_4-3*p_5^5, p_1*p_3+10*p_1-p_2*p_5^2-9*p_4*p_5+4*p_4, p_1*p_4-p_0*p_5+10*p_2, p_2^2-9*p_0*p_5-p_2-4*p_4*p_5^5, p_2*p_3-p_0*p_5+5*p_2, p_2*p_4-9*p_1+p_4-4*p_5^5, p_3^2+5*p_3-p_4*p_5^2-9*p_5, p_3*p_4-p_1-5*p_4, p_4^2-p_2}
      9
(29, x )
29
{p_0^2+4*p_0+13*p_1*p_5^3+2*p_2*p_5^5+6*p_2*p_5^2-11*p_3*p_5^5+12*p_4*p_5^4-5*p_4*p_5^3+7*p_4*p_5+p_5^5, p_0*p_1-9*p_0*p_5^2+4*p_1+p_2*p_5-13*p_3*p_5^4+2*p_4*p_5^6+12*p_5^5-12*p_5^4, p_0*p_2+2*p_1*p_5^4+6*p_1*p_5+4*p_2-3*p_4*p_5^4+10*p_4*p_5-11*p_5^6, p_0*p_3-9*p_1*p_5-6*p_2+14*p_4*p_5+2*p_5^6, p_0*p_4-9*p_2*p_5+2*p_3*p_5^4+4*p_4+13*p_5^4, p_1^2+5*p_0*p_5-9*p_1*p_5^2-6*p_2*p_5+5*p_2+14*p_4*p_5^2+2*p_5^7, p_1*p_2+p_1-9*p_2*p_5^2+2*p_3*p_5^5+4*p_4*p_5-8*p_4-13*p_5^5, p_1*p_3+13*p_1-p_2*p_5^2-6*p_4*p_5+12*p_4, p_1*p_4-p_0*p_5+13*p_2, p_2^2-9*p_0*p_5-14*p_2+2*p_4*p_5^5, p_2*p_3-p_0*p_5-8*p_2, p_2*p_4-9*p_1+14*p_4+2*p_5^5, p_3^2-8*p_3-p_4*p_5^2-6*p_5, p_3*p_4-p_1+8*p_4, p_4^2-p_2}
      9
(31, x )
31
{p_0^2+3*p_0-3*p_1*p_5^3+4*p_2*p_5^5+12*p_2*p_5^2+5*p_3*p_5^5+9*p_4*p_5^4-11*p_4*p_5^3-4*p_4*p_5+4*p_5^5, p_0*p_1-9*p_0*p_5^2+3*p_1+2*p_2*p_5+3*p_3*p_5^4+4*p_4*p_5^6+9*p_5^5-10*p_5^4, p_0*p_2+4*p_1*p_5^4+12*p_1*p_5+3*p_2-6*p_4*p_5^4-9*p_4*p_5+5*p_5^6, p_0*p_3-9*p_1*p_5-10*p_2-p_4*p_5+4*p_5^6, p_0*p_4-9*p_2*p_5+4*p_3*p_5^4+3*p_4-3*p_5^4, p_1^2-10*p_0*p_5-9*p_1*p_5^2-10*p_2*p_5-10*p_2-p_4*p_5^2+4*p_5^7, p_1*p_2+2*p_1-9*p_2*p_5^2+4*p_3*p_5^5+3*p_4*p_5+14*p_4+3*p_5^5, p_1*p_3+14*p_1-p_2*p_5^2-10*p_4*p_5+5*p_4, p_1*p_4-p_0*p_5+14*p_2, p_2^2-9*p_0*p_5+p_2+4*p_4*p_5^5, p_2*p_3-p_0*p_5+7*p_2, p_2*p_4-9*p_1-p_4+4*p_5^5, p_3^2+7*p_3-p_4*p_5^2-10*p_5, p_3*p_4-p_1-7*p_4, p_4^2-p_2}
      9
(37, x )
37
{p_0^2+16*p_0+11*p_1*p_5^3+10*p_2*p_5^5-7*p_2*p_5^2+16*p_3*p_5^5+11*p_4*p_5^4+2*p_4*p_5^3-4*p_4*p_5-12*p_5^5, p_0*p_1-9*p_0*p_5^2+16*p_1+5*p_2*p_5-11*p_3*p_5^4+10*p_4*p_5^6+11*p_5^5-p_5^4, p_0*p_2+10*p_1*p_5^4-7*p_1*p_5+16*p_2-15*p_4*p_5^4-4*p_4*p_5+16*p_5^6, p_0*p_3-9*p_1*p_5+10*p_2+16*p_4*p_5+10*p_5^6, p_0*p_4-9*p_2*p_5+10*p_3*p_5^4+16*p_4+11*p_5^4, p_1^2+7*p_0*p_5-9*p_1*p_5^2+10*p_2*p_5+7*p_2+16*p_4*p_5^2+10*p_5^7, p_1*p_2+5*p_1-9*p_2*p_5^2+10*p_3*p_5^5+16*p_4*p_5-13*p_4-11*p_5^5, p_1*p_3+17*p_1-p_2*p_5^2+10*p_4*p_5+15*p_4, p_1*p_4-p_0*p_5+17*p_2, p_2^2-9*p_0*p_5-16*p_2+10*p_4*p_5^5, p_2*p_3-p_0*p_5-10*p_2, p_2*p_4-9*p_1+16*p_4+10*p_5^5, p_3^2-10*p_3-p_4*p_5^2+10*p_5, p_3*p_4-p_1+10*p_4, p_4^2-p_2}

        5   2 3   3   3   15 4     5   4    3 3    30 2 2    9 2    45   2  
o4 = ({x , y x  - -y*x  - --x , y*x , y x - -y x - --y x  + --y x + --y*x  +
                  4       17                2      17       16      34      
     ------------------------------------------------------------------------
     225 3   3 3   15   4    9   3   45 4   5   9 4   30 3    27 3   45 2   
     ---x , y x  - --y*x  - --y*x  - --x , y  - -y  - --y x + --y  + --y x +
     289           17       16       68         4     17      16     17     
     ------------------------------------------------------------------------
     225   2   27 2   135       675 2     2   135     81   3        5  
     ---y*x  - --y  - ---y*x - ----x }, {p  - ---p  + --p p  - 27p p  -
     289       64     136      1156       0    17 0    4 1 5      2 5  
     ------------------------------------------------------------------------
          2         5   405   4   243   3   1215       729 5             2  
     81p p  - 243p p  - ---p p  - ---p p  - ----p p  + ---p , p p  - 9p p  -
        2 5       3 5    17 4 5    8  4 5    17  4 5    4  5   0 1     0 5  
     ------------------------------------------------------------------------
     135     27       81   4        6   405 5   243 4              4         
     ---p  - --p p  - --p p  - 27p p  - ---p  + ---p , p p  - 27p p  - 81p p 
      17 1    2 2 5    4 3 5      4 5    17 5    16 5   0 2      1 5      1 5
     ------------------------------------------------------------------------
       135     81   4   243           6                 15     27          6 
     - ---p  + --p p  + ---p p  - 243p , p p  - 9p p  - --p  + --p p  - 27p ,
        17 2    2 4 5    4  4 5       5   0 3     1 5   17 2    4 4 5      5 
     ------------------------------------------------------------------------
                         4   135     81 4   2   9           2   15       9  
     p p  - 9p p  - 27p p  - ---p  + --p , p  - -p p  - 9p p  - --p p  - -p 
      0 4     2 5      3 5    17 4    4 5   1   4 0 5     1 5   17 2 5   4 2
     ------------------------------------------------------------------------
       27   2      7         27         2        5   135       81     81 5 
     + --p p  - 27p , p p  - --p  - 9p p  - 27p p  - ---p p  + --p  - --p ,
        4 4 5      5   1 2    2 1     2 5      3 5    17 4 5    8 4    4 5 
     ------------------------------------------------------------------------
            3        2   15       9                  3     2           27    
     p p  - -p  - p p  - --p p  + -p , p p  - p p  - -p , p  - 9p p  - --p  -
      1 3   2 1    2 5   17 4 5   8 4   1 4    0 5   2 2   2     0 5    4 2  
     ------------------------------------------------------------------------
          5                3                 27        5   2   3        2  
     27p p , p p  - p p  - -p , p p  - 9p  + --p  - 27p , p  - -p  - p p  -
        4 5   2 3    0 5   4 2   2 4     1    4 4      5   3   4 3    4 5  
     ------------------------------------------------------------------------
     15                3     2
     --p , p p  - p  + -p , p  - p }, icR, | 25 21 20 11 10 6 |)
     17 5   3 4    1   4 4   4    2

o4 : Sequence

This is just like qthIntegralClosure except over QQ instead of ZZ/q. It calls qthIntegralClosure for several small primes q and reconciles the results using the Chinese Remainder Theorem and the extended Euclidean algorithm.

Ways to use rationalIntegralClosure :

rationalIntegralClosure(Matrix,Ring,List)

For the programmer

The object rationalIntegralClosure is a method function.

rationalIntegralClosure -- computes integral closures over the rationals

Synopsis

Description

See also

Ways to use rationalIntegralClosure :

For the programmer