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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 .00349142)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00010384)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00553816)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00929242)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0141844)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00655988)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0051925)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0052438)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00095842)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00071174)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007022)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00446136)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00523876)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00675052)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00693644)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0045348)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00618724)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00515312)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00568638)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00601484)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002712)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000787)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002176)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002504)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007388)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002434)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0031798)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000744)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00006036)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0005598)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00051848)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00198996)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00232362)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00040764)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00029058)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00067478)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00068086)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00260674)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00296896)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002194)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002254)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00003212)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .0000308)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0134364
#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 .00351392)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00010312)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00549046)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0093523)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0141577)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00661798)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00523864)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00524906)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00094522)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00070386)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00067446)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00456902)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00518176)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0570256)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00728128)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00465196)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0062468)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00521436)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00581188)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0061341)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002712)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009568)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023)   #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002188)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007814)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002516)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00323502)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007324)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00006394)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00057312)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0005198)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0020157)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00232904)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00039994)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00032878)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0006561)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00063504)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0026268)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00292594)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002142)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000245)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0130827)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0119716)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00061886)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00060626)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .0001332)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00014344)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002652)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002516)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0137782
#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 :