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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 .004448)   #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00012926)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00682596)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0119233)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .017981)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00848892)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00680948)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00680642)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00121048)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00082352)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0008587)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00583586)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00659504)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00872132)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00888756)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00588092)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0080473)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00655592)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00730692)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0077695)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002664)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008984)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000218)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003004)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011122)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002468)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0039766)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00010432)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000681)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0006806)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00062988)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00250306)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0029405)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00052796)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00036172)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00080636)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00076264)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00333724)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00375528)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000452)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000522)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00003576)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00004016)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0169071
#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 .0044757)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00012862)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00682586)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0119109)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0180175)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00848226)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00673292)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0066325)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00117648)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00088206)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00088814)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00579162)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00671134)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00871446)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00890572)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00587438)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00803306)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00670112)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00736272)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0077641)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002632)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00014086)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000233)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000284)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009958)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002146)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00491776)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001102)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007074)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00073662)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00066004)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00272728)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00302342)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00048958)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00036738)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00084492)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00082064)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00339184)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0038089)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002602)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002898)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0161525)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .015066)   #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00083624)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000798)   #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00016938)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00015124)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002836)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003046)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0168962
#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 :