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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 .00193389)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000055088)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00327368)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00528173)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00824)    #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0035553)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00280324)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00296744)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000567502)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000372208)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000374786)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00242532)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00291053)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00379991)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00390248)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00253845)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00338018)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0028044)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00310954)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00331028)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012794)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003503)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010312)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001017)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032586)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010146)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00166092)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033012)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038568)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000348624)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000328048)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00107407)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0012822)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000212168)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00016262)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000361124)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000363474)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00141825)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00160377)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010174)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001263)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000018554)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00001757)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00837064
#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 .0191285)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000070828)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00366001)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00529235)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00823402)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0036141)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00285628)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00298608)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000579588)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00037585)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000374666)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0024974)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00298504)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00381657)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00400666)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0024802)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00346947)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00282175)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00310931)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0033227)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011994)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035086)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001123)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001021)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003491)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011442)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00171668)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035182)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034882)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000351426)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000327426)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00108394)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00130361)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000216672)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000164586)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000374354)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000359248)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00144222)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00160466)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001069)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010258)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00721497)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00643773)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000272378)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000272402)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00008409)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000081958)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012584)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010922)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00848998
#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 :