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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 .0010242)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028752)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00160866)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0026959)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00419852)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00186994)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00149712)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00154885)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00029821)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000209784)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000202083)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00141342)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00159662)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00204398)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00212816)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00136154)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00186429)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156268)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00173635)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00182853)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008265)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021733)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004736)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005232)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019437)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005933)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000899825)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022556)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017224)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000177596)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000165673)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000594052)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000709676)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000113298)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000092639)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000203657)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000193653)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000798504)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000906214)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006068)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006771)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000009948)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00000886)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00443617
#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 .00108124)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031265)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00171381)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00285632)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00441519)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00199896)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00155402)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0015991)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000304311)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000211839)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000216008)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00137007)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0015993)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00208355)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00214464)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00133621)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00179036)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00147365)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00163841)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00176407)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005545)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018163)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005742)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005124)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001811)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004913)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000845087)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019424)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016918)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000167917)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000160319)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000569484)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000660407)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00010894)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000084915)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000204706)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000192705)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00074304)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000844525)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005188)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005196)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00352432)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00324966)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000140208)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000134893)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000038005)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000035726)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005575)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005808)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00419386
#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 :