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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 .0036191)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00013188)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0063888)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0105254)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0165122)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00751808)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0058209)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00593674)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00101906)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00074828)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00072248)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00531016)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00596126)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00764086)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00762142)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00500092)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0069678)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00573954)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0062608)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00666456)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003134)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011408)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003104)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003894)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011358)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002938)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0037232)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011136)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007794)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00066664)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00056238)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00234278)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00278328)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00043488)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00036628)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0007433)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00065956)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00295874)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00316474)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00005402)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003182)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00003716)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .0000418)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0146311
#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 .00372434)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001212)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00611902)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .010564)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0161743)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00739286)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00598944)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0057886)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00106652)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00085248)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00085512)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00528868)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00571378)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00795626)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0702051)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00506216)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00675254)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00564046)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00615398)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00650658)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003216)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00010976)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002322)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004042)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00012474)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00006816)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0037994)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011884)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000791)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0006928)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00055586)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00234744)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00255324)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00042942)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0003429)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00076558)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00067694)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00289242)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00318802)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003046)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003444)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .014294)   #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0130038)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00070914)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00066972)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00014964)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .0001338)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003476)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003672)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0149747
#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 :