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MinimalPrimes :: minprimes

minprimes -- minimal primes in a polynomial ring over a field

Synopsis

Description

Given an ideal in a polynomial ring, or a quotient of a polynomial ring whose base ring is either QQ or ZZ/p, return a list of minimal primes of the ideal.

i1 : R = ZZ/32003[a..e]

o1 = R

o1 : PolynomialRing
i2 : I = ideal"a2b-c3,abd-c2e,ade-ce2"

             2     3           2              2
o2 = ideal (a b - c , a*b*d - c e, a*d*e - c*e )

o2 : Ideal of R
i3 : C = minprimes I;
i4 : netList C

     +---------------------------+
o4 = |ideal (c, a)               |
     +---------------------------+
     |              2     3      |
     |ideal (e, d, a b - c )     |
     +---------------------------+
     |ideal (e, c, b)            |
     +---------------------------+
     |ideal (d, c, b)            |
     +---------------------------+
     |ideal (d - e, b - c, a - c)|
     +---------------------------+
     |ideal (d + e, b - c, a + c)|
     +---------------------------+
i5 : C2 = minprimes(I, Strategy=>"NoBirational", Verbosity=>2)
  Strategy: Linear            (time .00731578)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000197767)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0107035)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0183806)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0276634)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0129737)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0102672)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0101834)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00183727)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00137239)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00133121)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00911964)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0104946)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .013798)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0141337)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0092857)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0127106)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0104431)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0115859)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .012247)   #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039066)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011918)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003604)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003494)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000146333)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035633)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0060326)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001497)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000114207)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00101752)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000967873)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00384133)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00447137)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000752327)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000540867)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0013337)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00130956)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00530411)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00589428)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000054853)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004978)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00005828)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00005524)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0261112
#minprimes=6 #computed=10

                                  2     3
o5 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b),
     ------------------------------------------------------------------------
     ideal (d - e, b - c, a - c), ideal (d + e, b - c, a + c)}

o5 : List
i6 : C1 = minprimes(I, Strategy=>"Birational", Verbosity=>2)
  Strategy: Linear            (time .00736067)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000194774)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0107113)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0183213)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0277805)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .012971)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0102363)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0102206)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00185694)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00133255)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00136223)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00910133)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0105009)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0137738)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0141637)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00930594)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0126737)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0105552)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0116646)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0122777)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004038)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000148066)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036053)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038447)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00013078)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036007)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00600463)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001444)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000119259)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00100867)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000973427)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00385846)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00449799)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000752608)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000529646)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00130569)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00131881)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00527308)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00590459)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037394)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035914)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0250174)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0232733)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00119023)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00112688)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000286153)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00026286)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040434)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038027)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0260894
#minprimes=6 #computed=10

                                  2     3
o6 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b),
     ------------------------------------------------------------------------
     ideal (d - e, b - c, a - c), ideal (d + e, b - c, a + c)}

o6 : List

Caveat

This will eventually be made to work over GF(q), and over other fields too.

Ways to use minprimes :