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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 .00435882)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00013014)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00667398)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0115453)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0174382)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00817362)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00647346)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00646616)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00114052)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00083624)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00086628)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00564446)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00634382)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00843232)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00869098)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00563962)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00770218)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00646916)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00728966)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0077161)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004838)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001124)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004136)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007074)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009054)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002504)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00379492)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000971)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008204)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00066514)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00058018)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00222422)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00251854)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00042256)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00036498)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00074308)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00069866)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00286056)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00319924)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000264)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003136)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00004424)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00004352)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0164262
#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 .00369274)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00012268)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .006039)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0100871)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0156134)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00721854)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00566584)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00576538)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0010461)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00075746)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007413)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00492622)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00556462)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00764638)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00772926)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00499142)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00681172)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00566464)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00630734)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00654686)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003496)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008896)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002462)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003062)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009386)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004656)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00368028)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009348)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000697)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00064932)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00055094)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00223048)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00256848)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00044964)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00033968)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0007626)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00070684)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00284606)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00324824)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002516)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003152)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0138456)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0129777)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00059394)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00057078)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00017092)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00013866)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003646)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00005744)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0165639
#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 :