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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 .00131455)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040848)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00240815)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00374038)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00584007)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00249192)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00197518)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00209287)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000422562)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000272294)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000272258)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00169977)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00199181)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00260185)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00269695)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00168643)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0023177)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00191564)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0021281)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00228086)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001075)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002529)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006744)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007028)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024338)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006764)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00117968)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022942)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024532)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000272118)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000247994)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000780582)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000921732)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000154794)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000121422)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00024835)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000240712)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000993824)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00111558)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008526)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000704)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000012842)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011892)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00491991
#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 .00132385)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039944)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00237992)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0037444)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00588187)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00253689)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00203954)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0021259)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000427884)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000274392)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000273946)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00170825)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207605)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0160321)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00272608)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00177559)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00231044)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00191602)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00214956)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00228406)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010282)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025052)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006794)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008388)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025188)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006814)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00116575)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025782)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023812)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000274488)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000257966)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00077198)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000914286)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000155022)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0001201)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000247808)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000236924)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000978744)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00112188)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008064)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007116)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .004923)   #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00448392)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000224528)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000216846)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000052882)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00004844)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009742)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007548)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00495056
#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 :